Fun with Logarithms

Why do we use logarithms?

Logarithms were invented to deal with this type of problem.

\(3^x=25\).

How would we solve this for x? We could guess and check multiple times or we could create a method to solve it. Logarithms are that method.

Definition of Logarithms

A logarithm can be defined this way.

Logarithmic Form & Exponential Form

\(\displaystyle \log_{b}m=x \,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\, b^{x}=m\)

So we can take our original \(3^x=25\) and rewrite it as \(\log_{3}25=x\).

Would you like to practice some?

\(\textbf{1)}\) Write \(\log_{b}\frac{1}{5}=25\) in exponential form.

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The answer is \( b^{25}=\frac{1}{5} \)

\(\textbf{2)}\) Write \(\log_{4}x=2\) in exponential form.

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The answer is \( 4^{2}=x \)

\(\textbf{3)}\) Write \(\log_{b}25=2\) in exponential form.

Show Answer
The answer is \( b^{2}=25 \)

\(\textbf{4)}\) Write \(\log_{3}9=x\) in exponential form.

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The answer is \( 3^{x}=9 \)

\(\textbf{5)}\) Write \(m^5=32\) in logarithmic form.

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The answer is \( \log_{m}32=5 \)

\(\textbf{6)}\) Write \(3^x=27\) in logarithmic form.

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The answer is \( \log_{3}27=x \)

\(\textbf{7)}\) Write \(2^4=x\) in logarithmic form.

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The answer is \( \log_{2}x=4 \)

\(\textbf{8)}\) Write \(x^{10}=100\) in logarithmic form.

Show Answer
The answer is \( \log_{x}100=10 \)

Evaluating Logarithms

Evaluating logarithms is one of the funnest things to do with logarithms.

For example, if you have

\(log_2{8}\),

you can set it equal to x

\(log_2{8}=x\),

then rewrite it in exponential form as

\(2^x=8\),

then rewrite it as

\(2^x=2^3\),

then you know

\(x=3\).

Would you like to practice some?

\(\textbf{9)}\) \(\log_{3}9\)
Show Answer
The answer is \(2\)
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\(\,\,\,\,\,\log_{3}9=x\)

\(\,\,\,\,\,3^x=9\)

\(\,\,\,\,\,3^x=3^{2}\)

\(\,\,\,\,\,x=2\)

\(\textbf{10)}\) \(\log_{3}3\)
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The answer is \(1\)
Show Work

\(\,\,\,\,\,\log_{3}3=x\)

\(\,\,\,\,\,3^x=3\)

\(\,\,\,\,\,3^x=3^{1}\)

\(\,\,\,\,\,x=1\)

\(\textbf{11)}\) \(\log_{2}8\)
Show Answer
The answer is \(3\)
Show Work

\(\,\,\,\,\,\log_{2}8=x\)

\(\,\,\,\,\,2^x=8\)

\(\,\,\,\,\,2^x=2^{3}\)

\(\,\,\,\,\,x=3\)

\(\textbf{12)}\) \(\log_{4}16\)
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The answer is \(2\)
Show Work

\(\,\,\,\,\,\log_{4}16=x\)

\(\,\,\,\,\,4^x=16\)

\(\,\,\,\,\,4^x=4^2\)

\(\,\,\,\,\,x=2\)

\(\textbf{13)}\) \(\log_{6}6\)
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The answer is \(1\)
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\(\,\,\,\,\,\log_{6}6=x\)

\(\,\,\,\,\,6^x=6\)

\(\,\,\,\,\,6^x=6^1\)

\(\,\,\,\,\,x=1\)

\(\textbf{14)}\) \(\log_{2}32\)
Show Answer
The answer is \(5\)
Show Work

\(\,\,\,\,\,\log_{2}32=x\)

\(\,\,\,\,\,2^x=32\)

\(\,\,\,\,\,2^x=2^5\)

\(\,\,\,\,\,x=5\)

\(\textbf{15)}\) \(\log_{6}1\)
Show Answer
The answer is \(0\)
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\(\,\,\,\,\,\log_{6}1=x\)

\(\,\,\,\,\,6^x=1\)

\(\,\,\,\,\,6^x=6^0\)

\(\,\,\,\,\,x=0\)

\(\textbf{16)}\) \(\log_{2}\frac{1}{2}\)
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The answer is \(-1\)
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\(\,\,\,\,\,\log_{2}\frac{1}{2}=x\)

\(\,\,\,\,\,2^x=\frac{1}{2}\)

\(\,\,\,\,\,2^x=2^{-1}\)

\(\,\,\,\,\,x=-1\)

\(\textbf{17)}\) \(\log_{2}\frac{1}{4}\)
Show Answer
The answer is \(-2\)
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\(\,\,\,\,\,\log_{2}\frac{1}{4}=x\)

\(\,\,\,\,\,2^x=\frac{1}{4}\)

\(\,\,\,\,\,2^x=2^{-2}\)

\(\,\,\,\,\,x=-2\)

\(\textbf{18)}\) \(\log_{4}2\)
Show Answer
The answer is \(\frac{1}{2}\)
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\(\,\,\,\,\,\log_{4}2=x\)

\(\,\,\,\,\,4^x=2\)

\(\,\,\,\,\,4^x=2^{1/2}\)

\(\,\,\,\,\,x=\frac{1}{2}\)

Condensing Logarithms

Condensing Logarithms Notes

Condensing Logarithms Example

Would you like to practice some?

\(\textbf{19)}\) Condense: \(\log_{5}(3) + \log_{5}(x)\)
Show Answer
The answer is \( \log_{5}(3x) \)

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\(\,\,\,\,\,\,\log_{b}(M) + \log_{b}(N) = \log_{b}(MN)\,\,\, \left(\text{Product Rule}\right)\)

\(\,\,\,\,\,\,\log_{5}(3) + \log_{5}(x) = \log_{5}(3x) \)

\(\textbf{20)}\) Condense: \(\ln(6) + \ln(y)\)
Show Answer
The answer is \( \ln(6y) \)

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\(\,\,\,\,\,\,\ln(M) + \ln(N) = \ln(MN)\,\,\, \left(\text{Product Rule}\right)\)

\(\,\,\,\,\,\,\ln(6) + \ln(y) = \ln(6y) \)

\(\textbf{21)}\) Condense: \(\log_{2}(x) – \log_{2}(4)\)
Show Answer
The answer is \( \log_{2}\left(\frac{x}{4}\right) \)

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\(\,\,\,\,\,\,\log_{b}(M) – \log_{b}(N) = \log_{b}\left(\frac{M}{N}\right)\,\,\, \left(\text{Quotient Rule}\right)\)

\(\,\,\,\,\,\,\log_{2}(x) – \log_{2}(4) = \log_{2}\left(\frac{x}{4}\right) \)

\(\textbf{22)}\) Condense: \(\ln(y) – \ln(10)\)
Show Answer
The answer is \( \ln\left(\frac{y}{10}\right) \)

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\(\,\,\,\,\,\,\ln(M) – \ln(N) = \ln\left(\frac{M}{N}\right)\,\,\, \left(\text{Quotient Rule}\right)\)

\(\,\,\,\,\,\,\ln(y) – \ln(10) = \ln\left(\frac{y}{10}\right) \)

\(\textbf{23)}\) Condense: \(4\log_{3}(x)\)
Show Answer
The answer is \( \log_{3}(x^4) \)

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\(\,\,\,\,\,\,p\log_{b}(M) = \log_{b}(M^p)\,\,\, \left(\text{Power Rule}\right)\)

\(\,\,\,\,\,\,4\log_{3}(x) = \log_{3}(x^4) \)

\(\textbf{24)}\) Condense: \(\frac{1}{3}\ln(y)\)
Show Answer
The answer is \( \ln(y^{\frac{1}{3}}) \)

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\(\,\,\,\,\,\,p\ln(M) = \ln(M^p)\,\,\, \left(\text{Power Rule}\right)\)

\(\,\,\,\,\,\,\frac{1}{3}\ln(y) = \ln(y^{\frac{1}{3}}) \)

\(\textbf{25)}\) Write as a single logarithmic expression.
\(2\log_{5}(2)+\frac{1}{2}\log_{5}(x+3)-\log_{5}(x) \)

Show Answer
The answer is \( \log_{5}\left(\frac{4\sqrt{x+3}}{x}\right) \)
Show Work

\(\,\,\,\,\,\,2\log_{5}(2)+\frac{1}{2}\log_{5}(x+3)-\log_{5}(x) \)

\(\,\,\,\,\,\,\log_{5}(2^2) + \log_{5}((x+3)^{\frac{1}{2}}) – \log_{5}(x) \)

\(\,\,\,\,\,\,\log_{5}(4) + \log_{5}\left((x+3)^{\frac{1}{2}}\right) – \log_{5}(x) \)

\(\,\,\,\,\,\,\log_{5}\left(\frac{4\sqrt{x+3}}{x}\right) \)

\(\textbf{26)}\) Write as a single logarithmic expression.
\(2\log_{b}(x)+\log_{b}(z)-5\log_{b}(y) \)

Show Answer
The answer is \( \displaystyle\log_{b}\left(\frac{x^2z}{y^5}\right) \)
Show Work

\(\,\,\,\,\,\,2\log_{b}(x)+\log_{b}(z)-5\log_{b}(y) \)

\(\,\,\,\,\,\,\log_{b}(x^2) + \log_{b}(z) – \log_{b}(y^5) \)

\(\,\,\,\,\,\,\log_{b}\left(x^2z\right) – \log_{b}(y^5) \)

\(\,\,\,\,\,\,\log_{b}\left(\frac{x^2z}{y^5}\right) \)

\(\textbf{27)}\) Write as a single logarithmic expression.
\(\frac{1}{3}\log_{5}(z)-5\log_{5}(y)-2 \)

Show Answer
The answer is \( \displaystyle\log_{5}\left(\frac{\sqrt[3]{z}}{25y^5}\right) \)
Show Work

\(\,\,\,\,\,\,\frac{1}{3}\log_{5}(z)-5\log_{5}(y)-2 \)

\(\,\,\,\,\,\,\log_{5}\left(z^{\frac{1}{3}}\right) – \log_{5}(y^5) – \log_{5}(25)\,\,\,\,\, \left(\text{note: } \log_{5}(25)=2\right) \)

\(\,\,\,\,\,\,\log_{5}\left(\frac{\sqrt[3]{z}}{25y^5}\right) \)

\(\textbf{28)}\) Write as a single logarithmic expression.
\(\log_{2}(b)+\frac{1}{2}\log_{2}(z)-5 \)

Show Answer
The answer is \( \log_{2}\left(\frac{b\sqrt{z}}{32}\right) \)
Show Work

\(\,\,\,\,\,\,\log_{2}(b)+\frac{1}{2}\log_{2}(z)-5 \)

\(\,\,\,\,\,\,\log_{2}(b) + \log_{2}(z^{\frac{1}{2}}) – \log_{2}(2^5) \)

\(\,\,\,\,\,\,\log_{2}(b) + \log_{2}(\sqrt{z}) – \log_{2}(2^5) \)

\(\,\,\,\,\,\,\log_{2}\left(b\sqrt{z}\right) – \log_{2}(32) \)

\(\,\,\,\,\,\,\log_{2}\left(\frac{b\sqrt{z}}{32}\right) \)

\(\textbf{29)}\) Write as a single logarithmic expression.
\(2\log_{5}(x)+5\log_{5}(2)-\frac{1}{2}\log_{5}(z) \)

Show Answer
The answer is \( \displaystyle\log_{5}\left(\frac{32x^2}{\sqrt{z}}\right) \)
Show Work

\(\,\,\,\,\,\,2\log_{5}(x)+5\log_{5}(2)-\frac{1}{2}\log_{5}(z) \)

\(\,\,\,\,\,\,\log_{5}(x^2) + \log_{5}(2^5) – \log_{5}(z^{\frac{1}{2}}) \)

\(\,\,\,\,\,\,\log_{5}(x^2) + \log_{5}(32) – \log_{5}(\sqrt{z}) \)

\(\,\,\,\,\,\,\log_{5}(32x^2) – \log_{5}(\sqrt{z}) \)

\(\,\,\,\,\,\,\log_{5}\left(\frac{32x^2}{\sqrt{z}}\right) \)

\(\textbf{30)}\) Write as a single logarithmic expression.
\(5\ln(x+2)-3\ln(y)-2\ln(z) \)

Show Answer
The answer is \( \displaystyle\ln\left(\frac{\left(x+2\right)^5}{y^3 z^2}\right) \)
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\(\,\,\,\,\,\,5\ln(x+2)-3\ln(y)-2\ln(z) \)

\(\,\,\,\,\,\,\ln((x+2)^5) – \ln(y^3) – \ln(z^2) \)

\(\,\,\,\,\,\,\ln\left(\frac{\left(x+2\right)^5}{y^3z^2}\right) \)

\(\textbf{31)}\) Write as a single logarithmic expression.
\(\frac{1}{4}\log(x)-8\log(z)+1 \)

Show Answer
The answer is \( \displaystyle\log\left(\frac{10\sqrt[4]{x}}{z^8}\right) \)
Show Work

\(\,\,\,\,\,\,\frac{1}{4}\log(x)-8\log(z)+1 \)

\(\,\,\,\,\,\,\log(x^{\frac{1}{4}}) – \log(z^8) + \log(10)\,\,\,\,\, \left(\text{note: } \log(10)=1\right)\)

\(\,\,\,\,\,\,\log(\sqrt[4]{x}) – \log(z^8) + \log(10) \)

\(\,\,\,\,\,\,\log\left(\frac{\sqrt[4]{x}}{z^8}\right) + \log(10) \)

\(\,\,\,\,\,\,\log\left(\frac{10\sqrt[4]{x}}{z^8}\right) \)

\(\textbf{32)}\) Simplify.
\(\log(8)+2\log(5)-\log(2) \)

Show Answer
The answer is \( 2 \)
Show Work

\(\,\,\,\,\,\,\log(8)+2\log(5)-\log(2) \)

\(\,\,\,\,\,\,\log(2^3) + \log(5^2) – \log(2) \)

\(\,\,\,\,\,\,3\log(2) + 2\log(5) – \log(2) \)

\(\,\,\,\,\,\,\log(2^{3-1}) + \log(25) \)

\(\,\,\,\,\,\,\log(4) + \log(25) \)

\(\,\,\,\,\,\,\log(4 \cdot 25) \)

\(\,\,\,\,\,\,\log(100) \)

\(\,\,\,\,\,\,2 \)

Expanding Logarithms

Expanding Logarithms Example

Expanding Logarithms Notes

Would you like to practice some?

\(\textbf{33)}\) Expand: \(\log_{3}(2x)\)
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The answer is \( \log_{3}(2) + \log_{3}(x) \)

Show Work

\(\,\,\,\,\,\,\log_{b}(MN) = \log_{b}(M) + \log_{b}(N)\,\,\, \left(\text{Product Rule}\right)\)

\(\,\,\,\,\,\,\log_{3}(2x) = \log_{3}(2) + \log_{3}(x) \)

\(\textbf{34)}\) Expand: \(\ln(7a)\)
Show Answer
The answer is \( \ln(7) + \ln(a) \)

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\(\,\,\,\,\,\,\ln(MN) = \ln(M) + \ln(N)\,\,\, \left(\text{Product Rule}\right)\)

\(\,\,\,\,\,\,\ln(7a) = \ln(7) + \ln(a) \)

\(\textbf{35)}\) Expand: \(\log_{2}\left(\frac{5}{x}\right)\)
Show Answer
The answer is \( \log_{2}(5) – \log_{2}(x) \)

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\(\,\,\,\,\,\,\log_{b}\left(\frac{M}{N}\right) = \log_{b}(M) – \log_{b}(N)\,\,\, \left(\text{Quotient Rule}\right)\)

\(\,\,\,\,\,\,\log_{2}\left(\frac{5}{x}\right) = \log_{2}(5) – \log_{2}(x) \)

\(\textbf{36)}\) Expand: \(\ln\left(\frac{y}{3}\right)\)
Show Answer
The answer is \( \ln(y) – \ln(3) \)

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\(\,\,\,\,\,\,\ln\left(\frac{M}{N}\right) = \ln(M) – \ln(N)\,\,\, \left(\text{Quotient Rule}\right)\)

\(\,\,\,\,\,\,\ln\left(\frac{y}{3}\right) = \ln(y) – \ln(3) \)

\(\textbf{37)}\) Expand: \(\log_{4}(x^2)\)
Show Answer
The answer is \( 2\log_{4}(x) \)

Show Work

\(\,\,\,\,\,\,\log_{b}(M^p) = p\log_{b}(M)\,\,\, \left(\text{Power Rule}\right)\)

\(\,\,\,\,\,\,\log_{4}(x^2) = 2\log_{4}(x) \)

\(\textbf{38)}\) Expand: \(\ln(y^3)\)
Show Answer
The answer is \( 3\ln(y) \)

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\(\,\,\,\,\,\,\ln(M^p) = p\ln(M)\,\,\, \left(\text{Power Rule}\right)\)

\(\,\,\,\,\,\,\ln(y^3) = 3\ln(y) \)

\(\textbf{39)}\) \( \displaystyle\log_{b}\left(\frac{x^2z}{y^5}\right) \)

Show Answer
The answer is \( 2\log_{b}(x)+\log_{b}(z)-5\log_{b}(y) \)

\(\textbf{40)}\) \( \displaystyle\log_{5}\left(\frac{\sqrt[3]{z}}{25y^5}\right) \)

Show Answer
The answer is \( \frac{1}{3}\log_{5}(z)-5\log_{5}(y)-2 \)

\(\textbf{41)}\) \( \displaystyle\log_{2}\left(\frac{b\sqrt{n}}{32}\right) \)

Show Answer
The answer is \( \log_{2}(b)+\frac{1}{2}\log_{2}(n)-5 \)

\(\textbf{42)}\) \( \log_{5}\left(\frac{4\sqrt{x+3}}{x}\right) \)

Show Answer
The answer is \( \displaystyle2\log_{5}(2)+\frac{1}{2}\log_{5}(x+3)-\log_{5}(x) \)
Show Work

\(\,\,\,\,\,\,\displaystyle\log_{5}\left(\frac{4\sqrt{x+3}}{x}\right) \)

\(\,\,\,\,\,\,\displaystyle\log_{5}\left(4\right) + \log_{5}\left(\sqrt{x+3}\right) – \log_{5}\left(x\right)\)

\(\,\,\,\,\,\,\displaystyle\log_{5}\left(2\right)^2 + \log_{5}(x+3)^{1/2} – \log_{5}\left(x\right)\)

\(\,\,\,\,\,\,2\log_{5}\left(2\right) + \frac{1}{2} \log_{5}(x+3) – \log_{5}\left(x\right)\)

Solving Logarithmic Equations

Lesson

Practice Problems & Videos

Solve for x.

\(\textbf{43)}\) \(\log_{3}x=2\)
Show Answer
\(x=9\)
Show Work

\(\,\,\,\,\,\,\log_{b}x=c \,\,\Rightarrow\,\, b^c=x\)

\(\,\,\,\,\,\,\log_{3}x=2 \,\,\Rightarrow \,\,3^2=x\)

\(\,\,\,\,\,\,x=9\)

\(\textbf{44)}\) \(\log_{2}(4x)=3\)
Show Answer
\(x=2\)
Show Work

\(\,\,\,\,\,\,\log_{b}x=c \,\,\Rightarrow\,\, b^c=x\)

\(\,\,\,\,\,\,\log_{2}(4x)=3 \,\,\Rightarrow \,\,2^3=4x\)

\(\,\,\,\,\,\,8=4x\)

\(\,\,\,\,\,\,2=x\)

\(\textbf{45)}\) \(\log_{4}(3x+1)=2\)
Show Answer
\(x=5\)
Show Work

\(\,\,\,\,\,\,\log_{b}x=c \,\,\Rightarrow\,\, b^c=x\)

\(\,\,\,\,\,\,\log_{4}(3x+1)=2 \,\,\Rightarrow \,\,4^2=3x+1\)

\(\,\,\,\,\,\,16=3x+1\)

\(\,\,\,\,\,\,15=3x\)

\(\,\,\,\,\,\,5=x\)

\(\textbf{46)}\) \(\log_{2}(5x)=\log_{2}(2x+12)\)
Show Answer
\(x=4\)
Show Work

\(\,\,\,\,\,\,\log_{2}(5x)=\log_{2}(2x+12)\)

\(\,\,\,\,\,\,5x=2x+12\)

\(\,\,\,\,\,\,3x=12\)

\(\,\,\,\,\,\,x=4\)

\(\textbf{47)}\) \(\log_{5}(x+1)+\log_{5}(4)=\log_{5}(24)\)
Show Answer
\(x=5\)
Show Work

\(\,\,\,\,\,\,\log_{5}(x+1)+\log_{5}(4)=\log_{5}(24)\)

\(\,\,\,\,\,\,\log_{5}(x+1)(4)=\log_{5}(24)\)

\(\,\,\,\,\,\,(x+1)(4)=(24)\)

\(\,\,\,\,\,\,4x+4=24\)

\(\,\,\,\,\,\,4x=20\)

\(\,\,\,\,\,\,x=5\)

\(\textbf{48)}\) \(\log_{6}(2x)+\log_{6}(x-1)=\log_{6}(10x)\)
Show Answer
\(x=6\)
Show Work

\(\,\,\,\,\,\,\log_{6}(2x)+\log_{6}(x-1)=\log_{6}(10x)\)

\(\,\,\,\,\,\,\log_{6}(2x)(x-1)=\log_{6}(10x)\)

\(\,\,\,\,\,\,(2x)(x-1)=(10x)\)

\(\,\,\,\,\,\,2x^2-2x=10x\)

\(\,\,\,\,\,\,2x^2-12x=0\)

\(\,\,\,\,\,\,x^2-6x=0\)

\(\,\,\,\,\,\,x(x-6)=0\)

\(\,\,\,\,\,\,x=0 \text{ or } x=6\)

\(\,\,\,\,\,\,x=0 \text{ is extraneous }\)

\(\,\,\,\,\,\,x=6\)

\(\textbf{49)}\) \(\log_{3}(4)-\log_{3}(x)=\log_{3}(2x-2)-\log_{3}(3x+4)\)
Show Answer
\(x=8\)
Show Work

\(\,\,\,\,\,\,\log_{3}(4)-\log_{3}(x)=\log_{3}(2x-2)-\log_{3}(3x+4)\)

\(\,\,\,\,\,\,\log_{3}\left(\frac{4}{x}\right)=\log_{3}\left(\frac{2x-2}{3x+4}\right)\)

\(\,\,\,\,\,\,\displaystyle\frac{4}{x}=\frac{2x-2}{3x+4}\)

\(\,\,\,\,\,\,(4)(3x+4)=(x)(2x-2)\)

\(\,\,\,\,\,\,12x+16=2x^2-2x\)

\(\,\,\,\,\,\,0=2x^2-14x-16\)

\(\,\,\,\,\,\,0=x^2-7x-8\)

\(\,\,\,\,\,\,0=(x-8)(x+1)\)

\(\,\,\,\,\,\,x=-1 \text{ or } x=8\)

\(\,\,\,\,\,\,x=-1 \text{ is extraneous}\)

\(\,\,\,\,\,\,x=8\)

\(\textbf{50)}\) \(\frac{1}{2}\log_{}(x^4)-\log_{}(2x-1)=\log_{}(x+2)-\log_{}(3)\)
Show Answer
\(x=1 \text{ or }x=2\)
Show Work

\(\,\,\,\,\,\,\frac{1}{2}\log_{}(x^4)-\log_{}(2x-1)=\log_{}(x+2)-\log_{}(3)\)

\(\,\,\,\,\,\,\log_{}\left(\sqrt{x^4}\right)-\log_{}(2x-1)=\log_{}(x+2)-\log_{}(3)\)

\(\,\,\,\,\,\,\log_{}\left(x^2\right)-\log_{}(2x-1)=\log_{}(x+2)-\log_{}(3)\)

\(\,\,\,\,\,\,\log_{}\left(\frac{x^2}{2x-1}\right)=\log_{}\left(\frac{x+2}{3}\right)\)

\(\,\,\,\,\,\,\displaystyle\frac{x^2}{2x-1}=\frac{x+2}{3}\)

\(\,\,\,\,\,\,3x^2=(x+2)(2x-1)\)

\(\,\,\,\,\,\,3x^2=2x^2+3x-2\)

\(\,\,\,\,\,\,x^2-3x+2=0\)

\(\,\,\,\,\,\,(x-1)(x-2)=0\)

\(\,\,\,\,\,\,x=1 \text{ or } x=2\)

\(\textbf{51)}\) \(\ln_{}(10x)=\ln_{}(4x+1)+\ln_{}(3-x)\)
Show Answer
\(x=1\)
Show Work

\(\,\,\,\,\,\ln_{}(10x)=\ln_{}(4x+1)+\ln_{}(3-x)\)

\(\,\,\,\,\,\ln_{}(10x)=\ln_{}(4x+1)(3-x)\)

\(\,\,\,\,\,10x=(4x+1)(3-x)\)

\(\,\,\,\,\,10x=-4x^2+11x+3\)

\(\,\,\,\,\,4x^2-x-3=0\)

\(\,\,\,\,\,(4x+3)(x-1)=0\)

\(\,\,\,\,\,x=-\frac{3}{4} \text{ or } x=1\)

\(\,\,\,\,\,x=-\frac{3}{4} \text{ is extraneous}\)

\(\,\,\,\,\,x=x=1\)

\(\textbf{52)}\) \(\log_{6}(8x-4)=2\)
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\(x=5\)
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\(\,\,\,\,\,\,\log_{b}x=c \,\,\Rightarrow\,\, b^c=x\)

\(\,\,\,\,\,\,\log_{6}(8x-4)=2 \,\,\Rightarrow \,\,6^2=8x-4\)

\(\,\,\,\,\,\,36=8x-4\)

\(\,\,\,\,\,\,40=8x\)

\(\,\,\,\,\,\,5=x\)

\(\textbf{53)}\) \(\log_{7}(6x-5)=2\)
Show Answer
\(x=9\)
Show Work

\(\,\,\,\,\,\,\log_{b}x=c \,\,\Rightarrow\,\, b^c=x\)

\(\,\,\,\,\,\,\log_{7}(6x-5)=2 \,\,\Rightarrow \,\,7^2=6x-5\)

\(\,\,\,\,\,\,49=6x-5\)

\(\,\,\,\,\,\,54=6x\)

\(\,\,\,\,\,\,9=x\)

\(\textbf{54)}\) \(\log_{}(x-1)=\log_{}(5x)-1\)
Show Answer
\(x=2\)
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\(\,\,\,\,\,\,\log_{}(x-1)=\log_{}(5x)-1\)

\(\,\,\,\,\,\,\log_{}(x-1)=\log_{}(5x)-\log_{}(10)\)

\(\,\,\,\,\,\,\log_{}(x-1)=\log_{}\left(\frac{5x}{10}\right)\)

\(\,\,\,\,\,\,x-1=\displaystyle\frac{5x}{10}\)

\(\,\,\,\,\,\,x-1=\displaystyle\frac{x}{2}\)

\(\,\,\,\,\,\,2(x-1)=x\)

\(\,\,\,\,\,\,2x-2=x\)

\(\,\,\,\,\,\,x-2=0\)

\(\,\,\,\,\,\,x=2\)

\(\textbf{55)}\) \(\log_{4}(3x-10)=\log_{4}(8x)-2\)
Show Answer
\(x=4\)
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\(\,\,\,\,\,\,\log_{4}(3x-10)=\log_{4}(8x)-2\)

\(\,\,\,\,\,\,\log_{4}(3x-10)=\log_{4}(8x)-\log_{4}(16)\)

\(\,\,\,\,\,\,\log_{4}(3x-10)=\log_{4}\left(\frac{8x}{16}\right)\)

\(\,\,\,\,\,\,3x-10=\displaystyle\frac{8x}{16}\)

\(\,\,\,\,\,\,3x-10=\displaystyle\frac{x}{2}\)

\(\,\,\,\,\,\,2(3x-10)=x\)

\(\,\,\,\,\,\,6x-20=x\)

\(\,\,\,\,\,\,5x-20=0\)

\(\,\,\,\,\,\,5x=20\)

\(\,\,\,\,\,\,x=4\)

\(\textbf{56)}\) \(\log_{3}(x^2-2x)=\log_{3}(2-x)+2\)
Show Answer
\(x=-9\)
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\(\,\,\,\,\,\,\log_{3}(x^2-2x)=\log_{3}(2-x)+2\)

\(\,\,\,\,\,\,\log_{3}(x^2-2x)=\log_{3}(2-x)+\log_{3}(9)\)

\(\,\,\,\,\,\,\log_{3}(x^2-2x)=\log_{3}(2-x)(9)\)

\(\,\,\,\,\,\,x^2-2x=(2-x)(9)\)

\(\,\,\,\,\,\,x^2-2x=18-9x\)

\(\,\,\,\,\,\,x^2+7x-18=0\)

\(\,\,\,\,\,\,(x+9)(x-2)=0\)

\(\,\,\,\,\,\,x=-9 \text{ or } x=2\)

\(\,\,\,\,\,\,x=2 \text{ is extraneous }\)

\(\,\,\,\,\,\,x=-9\)

\(\textbf{57)}\) \(\log_{2}(x+1)+\log_{2}(x-1)=\log_{2}(2x+2)+1\)
Show Answer
\(\text{The answer is }x=5\)
Show Work

\(\,\,\,\,\,\,\log_{2}(x+1)+\log_{2}(x-1)=\log_{2}(2x+2)+1\)

\(\,\,\,\,\,\,\log_{2}(x+1)+\log_{2}(x-1)=\log_{2}(2x+2)+\log_{2}2\)

\(\,\,\,\,\,\,\log_{2}(x+1)(x-1)=\log_{2}(2x+2)(2)\)

\(\,\,\,\,\,\,(x+1)(x-1)=(2x+2)(2)\)

\(\,\,\,\,\,\,x^2-1=4x+4\)

\(\,\,\,\,\,\,x^2-4x-5=0\)

\(\,\,\,\,\,\,(x-5)(x+1)=0\)

\(\,\,\,\,\,\,x=-1 \,\text{ or }\, x=5\)

\(\,\,\,\,\,\,x=-1 \text{ is extraneous}\)

\(\,\,\,\,\,\,\text{The answer is }x=5\)

Change of Base Formula

The change of base formula allows you to rewrite logarithms in terms of logarithms with different bases, often using base 10 (\(\log\)) or base \(e\) (\(\ln\)). It is given by:

\(\log_{b}x = \frac{\log{x}}{\log{b}} \,\, \text{or} \,\, \log_{b}x = \frac{\ln{x}}{\ln{b}}\)

This is particularly useful for evaluating logarithms on calculators that may not directly support arbitrary bases. For example:

\(\log_{3}10 = \frac{\ln{10}}{\ln{3}} \approx 2.096\)

Practice Problems

\(\textbf{58)}\) Simplify \(\log_{4}{18}\) using the change of base formula.
Show Answer
The answer is \(\frac{\ln{18}}{\ln{4}} \approx 2.085\)

\(\textbf{59)}\) Simplify \(\log_{7}{49}\) using the change of base formula.
Show Answer
The answer is \(\frac{\ln{49}}{\ln{7}} = 2\)

\(\textbf{60)}\) Simplify \(\log_{5}{125}\) using the change of base formula.
Show Answer
The answer is \(\frac{\ln{125}}{\ln{5}} = 3\)

\(\textbf{61)}\) Simplify \(\log_{8}{32}\) using the change of base formula.
Show Answer
The answer is \(\frac{\ln{32}}{\ln{8}} \approx 1.667\)

\(\textbf{62)}\) Simplify \(\log_{2}{50}\) using the change of base formula.
Show Answer
The answer is \(\frac{\ln{50}}{\ln{2}} \approx 5.644\)

Natural Logarithms

Natural logarithms use the base \(e\), where \(e \approx 2.71828\). They are denoted as \(\ln\). For example:

\(\ln{x} \,\, \text{is equivalent to} \,\, \log_{e}x.\)

Natural logs are widely used in mathematics, they have applications in exponential growth and decay models.

Practice Problems

\(\textbf{63)}\) Solve \(\ln{(x+2)} = 3\).
Show Answer
The answer is \(x = e^3 – 2 \approx 18.085\)

Show Work
\(\,\,\,\,\,\,\ln{(x+2)} = 3\)

\(\,\,\,\,\,\,\text{Rewrite } \ln{(x+2)} \text{ as } \log_{e}{(x+2)}:\)

\(\,\,\,\,\,\,\log_{e}{(x+2)} = 3\)

\(\,\,\,\,\,\,\text{Rewrite using the definition of logarithms: } e^3 = x+2\)

\(\,\,\,\,\,\,x+2 = e^3\)

\(\,\,\,\,\,\,x = e^3 – 2\)

\(\,\,\,\,\,\,x \approx 18.085\)

\(\textbf{64)}\) Solve \(\ln{(3x)} = 5\).
Show Answer
The answer is \(x = \frac{e^5}{3} \approx 49.471\)

Show Work
\(\,\,\,\,\,\,\ln{(3x)} = 5\)

\(\,\,\,\,\,\,\text{Rewrite } \ln{(3x)} \text{ as } \log_{e}{(3x)}:\)

\(\,\,\,\,\,\,\log_{e}{(3x)} = 5\)

\(\,\,\,\,\,\,\text{Rewrite using the definition of logarithms: } e^5 = 3x\)

\(\,\,\,\,\,\,3x = e^5\)

\(\,\,\,\,\,\,x = \frac{e^5}{3}\)

\(\,\,\,\,\,\,x \approx 49.471\)

\(\textbf{65)}\) Solve \(\ln{(x-4)} = 2\).
Show Answer
The answer is \(x = e^2 + 4 \approx 11.389\)

Show Work
\(\,\,\,\,\,\,\ln{(x-4)} = 2\)

\(\,\,\,\,\,\,\text{Rewrite } \ln{(x-4)} \text{ as } \log_{e}{(x-4)}:\)

\(\,\,\,\,\,\,\log_{e}{(x-4)} = 2\)

\(\,\,\,\,\,\,\text{Rewrite using the definition of logarithms: } e^2 = x-4\)

\(\,\,\,\,\,\,x-4 = e^2\)

\(\,\,\,\,\,\,x = e^2 + 4\)

\(\,\,\,\,\,\,x \approx 11.389\)

\(\textbf{66)}\) Solve \(\ln{(5x-1)} = 4\).
Show Answer
The answer is \(x = \frac{e^4 + 1}{5} \approx 11.120\)

Show Work
\(\,\,\,\,\,\,\ln{(5x-1)} = 4\)

\(\,\,\,\,\,\,\text{Rewrite } \ln{(5x-1)} \text{ as } \log_{e}{(5x-1)}:\)

\(\,\,\,\,\,\,\log_{e}{(5x-1)} = 4\)

\(\,\,\,\,\,\,\text{Rewrite using the definition of logarithms: } e^4 = 5x-1\)

\(\,\,\,\,\,\,5x-1 = e^4\)

\(\,\,\,\,\,\,5x = e^4 + 1\)

\(\,\,\,\,\,\,x = \frac{e^4 + 1}{5}\)

\(\,\,\,\,\,\,x \approx 11.120\)

\(\textbf{67)}\) Solve \(\ln{(2x+3)} = 6\).
Show Answer
The answer is \(x = \frac{e^6 – 3}{2} \approx 200.214\)

Show Work
\(\,\,\,\,\,\,\ln{(2x+3)} = 6\)

\(\,\,\,\,\,\,\text{Rewrite } \ln{(2x+3)} \text{ as } \log_{e}{(2x+3)}:\)

\(\,\,\,\,\,\,\log_{e}{(2x+3)} = 6\)

\(\,\,\,\,\,\,\text{Rewrite using the definition of logarithms: } e^6 = 2x+3\)

\(\,\,\,\,\,\,2x+3 = e^6\)

\(\,\,\,\,\,\,2x = e^6 – 3\)

\(\,\,\,\,\,\,x = \frac{e^6 – 3}{2}\)

\(\,\,\,\,\,\,x \approx 200.214\)

Extraneous Solutions and Domain Restrictions

When solving logarithmic equations, extraneous solutions may arise. These are solutions that do not satisfy the domain of the original logarithmic expression. Remember:

The domain of a logarithmic function \(\log_{b}{x}\) is \(x > 0\).

Check all solutions to ensure they are within the domain.

See Related Pages\(\)

\(\bullet\text{ Evaluating Logarithms}\)
\(\,\,\,\,\,\,\,\,\log_{2}(8)…\)

\(\bullet\text{ Decibel Problems}\)
\(\,\,\,\,\,\,\,\,N_{dB}=10\log \left(\frac{P}{10^{-12}}\right)…\)

\(\bullet\text{ Earthquake Problems}\)
\(\,\,\,\,\,\,\,\,M=\log\frac{I}{10^{-4}}…\)

\(\bullet\text{ Domain and Range Logarithmic Functions}\)
\(\,\,\,\,\,\,\,\,f(x)=log(x) \rightarrow \text{Domain:} x\gt0… \)

\(\bullet\text{ Graphing Logarithmic Functions}\)
\(\,\,\,\,\,\,\,\,f(x)=log_{2}(x)\)

\(\bullet\text{ Solving Logarithmic Equations}\)
\(\,\,\,\,\,\,\,\,\log_{2}(5x)=\log_{2}(2x+12)…\)

\(\bullet\text{ Inverse of Logarithmic Functions}\)
\(\,\,\,\,\,\,\,\,f(x)=log_{2}(x) \rightarrow f^{-1}(x)=2^x\)

\(\bullet\text{ Half Life Questions}\)
\(\,\,\,\,\,\,\,\,A_t=A_0e^{kt}\)

\(\bullet\text{ Compound Interest}\)
\(\,\,\,\,\,\,\,\,A=P\left(1+\frac{r}{n} \right)^{nt}\,\,\,\,\,\,\,\,A=P e^{rt}\)

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