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Practice Problems & Videos
Solve for x.
\(\textbf{1)}\) \(5^{2x} = 5^{x+4}\)
The answer is \(x =4\)
\(\,\,\,\,\,\,5^{2x} = 5^{x+4}\)
\(\,\,\,\,\,\,2x = x+4\)
\(\,\,\,\,\,\,x = 4\)
\(\,\,\,\,\,\,\)The answer is \(x =4\)
\(\textbf{2)}\) \(6^{9 + 2x} = 6^{3 + 8x}\)
The answer is \(x =1\)
\(\,\,\,\,\,\,6^{9 + 2x} = 6^{3 + 8x}\)
\(\,\,\,\,\,\,9 + 2x = 3 + 8x\)
\(\,\,\,\,\,\,9 = 3 + 6x\)
\(\,\,\,\,\,\,6 = 6x\)
\(\,\,\,\,\,\,1 = x\)
\(\,\,\,\,\,\,\)The answer is \(x =1\)
\(\textbf{3)}\) \(3^{4 – 2x} = 3^{8 – x}\)
The answer is \(x=-4\)
\(\,\,\,\,\,\,3^{4 – 2x} = 3^{8 – x}\)
\(\,\,\,\,\,\,4 – 2x = 8 – x\)
\(\,\,\,\,\,\,4 = 8 + x\)
\(\,\,\,\,\,\,-4 = x\)
\(\,\,\,\,\,\,\)The answer is \(x =-4\)
\(\textbf{4)}\) \(4^{x^{2}} = 4^{5x-6}\)
The answer is \(x=2, \,\,\,x=3\)
\(\,\,\,\,\,\,4^{x^{2}} = 4^{5x-6}\)
\(\,\,\,\,\,\,x^{2} = 5x-6\)
\(\,\,\,\,\,\,x^{2}-5x+6=0\)
\(\,\,\,\,\,\,(x-2)(x-3)=0\)
\(\,\,\,\,\,\,x-2=0 \,\,\text{or} \,\,x-3=0\)
\(\,\,\,\,\,\,\)The answer is \(x=2, \,\,\text{or}\,\, x=3\)
\(\textbf{5)}\) \(7^{x^{2}} = 7^{x+12}\)
The answer is \(x=-3\,\,\,x=4\)
\(\,\,\,\,\,\,7^{x^{2}} = 7^{x+12}\)
\(\,\,\,\,\,\,x^{2} = x+12\)
\(\,\,\,\,\,\,x^{2}-x-12=0\)
\(\,\,\,\,\,\,(x-4)(x+3)=0\)
\(\,\,\,\,\,\,x-4=0 \,\,\text{or} \,\,x+3=0\)
\(\,\,\,\,\,\,\)The answer is \(x=4, \,\,\text{or}\,\, x=-3\)
\(\textbf{6)}\) \(4^{4x}=2^{2x+6} \)
The answer is \( x=1 \)
\(\textbf{7)}\) \( 3^{x}=\frac{1}{27} \)
The answer is \( x=-3 \)
\(\,\,\,\,\,\,3^{x}=\frac{1}{27}\)
\(\,\,\,\,\,\,3^{x}=\left(3^{-3}\right)\)
\(\,\,\,\,\,\,x=-3\)
\(\,\,\,\,\,\,\)The answer is \( x=-3 \)
\(\textbf{8)}\) \(\displaystyle 6^{2x+2} = \frac{1}{36^{3x +7}}\)
The answer is \(x =-2\)
\(\,\,\,\,\,\,\displaystyle 6^{2x+2} = \frac{1}{36^{3x +7}}\)
\(\,\,\,\,\,\,\displaystyle 6^{2x+2} = \left(\frac{1}{36}\right)^{3x +7}\)
\(\,\,\,\,\,\,\displaystyle 6^{2x+2} = \left(6^{-2}\right)^{3x +7}\)
\(\,\,\,\,\,\,\displaystyle 6^{2x+2} = \left(6\right)^{-6x -14}\)
\(\,\,\,\,\,\,\displaystyle 2x+2 = -6x -14\)
\(\,\,\,\,\,\,\displaystyle 8x+2 = -14\)
\(\,\,\,\,\,\,\displaystyle 8x = -16\)
\(\,\,\,\,\,\,\)The answer is \(x =-2\)
\(\textbf{9)}\) \(4^x = 8^{x-2}\)
The answer is \(x =6\)
\(\,\,\,\,\,4^x = 8^{x-2}\)
\(\,\,\,\,\,\left(2^2\right)^x=\left(2^3\right)^{x-2}\)
\(\,\,\,\,\,2^{2x}=2^{3x-6}\)
\(\,\,\,\,\,2x=3x-6\)
\(\,\,\,\,\,-1x=-6\)
\(\,\,\,\,\,\) The answer is \(x =6\)
\(\textbf{10)}\) \(6^{x + 1} = 1\)
The answer is \(x =-1\)
\(\,\,\,\,\,\,6^{x + 1} = 1\)
\(\,\,\,\,\,\,6^{x + 1} = 6^0\)
\(\,\,\,\,\,\,x + 1 = 0\)
\(\,\,\,\,\,\,x = -1\)
\(\,\,\,\,\,\,\)The answer is \(x =-1\)
\(\textbf{11)}\) \(5^{2x – 4} = 1\)
The answer is \(x =2\)
\(\,\,\,\,\,\,5^{2x – 4} = 1\)
\(\,\,\,\,\,\,5^{2x – 4} = 5^0\)
\(\,\,\,\,\,\,2x – 4 = 0\)
\(\,\,\,\,\,\,2x = 4\)
\(\,\,\,\,\,\,x = 2\)
\(\,\,\,\,\,\,\)The answer is \(x =2\)
Challenge Questions
Solve the following using logarithms
\(\textbf{12)}\) \(6 = 3^{2x+3}\)
The answer is \(x = \frac{\left[\frac{\ln{6}}{\ln{3}}-3\right]}{2} \approx -0.6845\)
\(\,\,\,\,\,\,6 = 3^{2x+3}\)
\(\,\,\,\,\,\,\ln{6} = \ln{3^{2x+3}}\)
\(\,\,\,\,\,\,\ln{6} = (2x+3)\ln{3}\)
\(\,\,\,\,\,\,\displaystyle \frac{\ln{6}}{\ln{3}} = 2x+3\)
\(\,\,\,\,\,\,\displaystyle \frac{\ln{6}}{\ln{3}}-3 = 2x\)
\(\,\,\,\,\,\,\displaystyle \frac{\left[\frac{\ln{6}}{\ln{3}}-3\right]}{2} = x\)
\(\,\,\,\,\,\,\)The answer is \(x \approx -0.6845\)
\(\textbf{13)}\) \(3^{x+5} = 5^{x+1}\)
The answer is \(x = \displaystyle \frac{ \ln{5}-5\ln{3}}{\ln{3}-\ln{5}} \approx 7.6026\)
\(\,\,\,\,\,\,3^{x+5} = 5^{x+1}\)
\(\,\,\,\,\,\,\ln{3^{x+5}} = \ln{5^{x+1}}\)
\(\,\,\,\,\,\,(x+5)\ln{3} = (x+1)\ln{5}\)
\(\,\,\,\,\,\,x\ln{3}+5\ln{3} = x\ln{5}+\ln{5}\)
\(\,\,\,\,\,\,x\ln{3}-x\ln{5}= \ln{5}-5\ln{3}\)
\(\,\,\,\,\,\,x\left(\ln{3}-\ln{5}\right)= \ln{5}-5\ln{3}\)
\(\,\,\,\,\,\,x= \displaystyle \frac{ \ln{5}-5\ln{3}}{\ln{3}-\ln{5}}\)
\(\,\,\,\,\,\,\)The answer is \(x \approx 7.6026\)
\(\textbf{14)}\) \(4^{4x} = 6\)
The answer is \(x = \displaystyle\frac{\ln{6}}{4\ln{4}} \approx 0.3231\)
\(\,\,\,\,\,\,4^{4x} = 6\)
\(\,\,\,\,\,\,\ln{4^{4x}} = \ln{6}\)
\(\,\,\,\,\,\,4x\ln{4} = \ln{6}\)
\(\,\,\,\,\,\,x = \displaystyle\frac{\ln{6}}{4\ln{4}}\)
\(\,\,\,\,\,\,\)The answer is \(x \approx 0.3231\)
\(\textbf{15)}\) \(e^x=40\)
The answer is \(x =\ln{40} \approx 3.6889\)
\(\,\,\,\,\,\,e^x=40\)
\(\,\,\,\,\,\,\ln{e^x}=\ln{40}\)
\(\,\,\,\,\,\,x\ln{e}=\ln{40}\)
\(\,\,\,\,\,\,x=\ln{40}\)
\(\,\,\,\,\,\,\)The answer is \(x \approx 3.6889\)
\(\textbf{16)}\) \(10^{x+2}-1=5\)
The answer is \(x=\log{6}-2 \approx -1.2218\)
\(\,\,\,\,\,\,10^{x+2}-1=5\)
\(\,\,\,\,\,\,10^{x+2}=6\)
\(\,\,\,\,\,\,\log{10^{x+2}}=\log{6}\)
\(\,\,\,\,\,\,(x+2)\log{10}=\log{6}\)
\(\,\,\,\,\,\,x+2=\log{6}\)
\(\,\,\,\,\,\,x=\log{6}-2\)
\(\,\,\,\,\,\,\)The answer is \(x=\log{6}-2 \approx -1.2218\)
\(\textbf{17)}\) \(e^{2x}-6e^x+8=0\)
The answer is \(x=\ln(2)\approx0.6931,\,\,\, x=\ln(4)\approx1.3863\)
\(\,\,\,\,\,\,e^{2x}-6e^x+8=0\)
\(\,\,\,\,\,\,\left(e^{x}-4\right)\left(e^{x}-2\right)=0\)
\(\,\,\,\,\,\,e^{x}-4=0\,\,\text{or}\,\,e^{x}-2=0\)
\(\,\,\,\,\,\,e^{x}=4\,\,\text{or}\,\,e^{x}=2\)
\(\,\,\,\,\,\,\ln{e^{x}}=\ln{4}\,\,\text{or}\,\,\ln{e^{x}}=\ln{2}\)
\(\,\,\,\,\,\,x=\ln{4}\,\,\text{or}\,\,x=\ln{2}\)
\(\,\,\,\,\,\,\)The answer is \(x=\ln(2)\approx0.6931,\,\,\, x=\ln(4)\approx1.3863\)
\(\textbf{18)}\) \(e^{2x}+6e^x+8=0\)
The answer is No Solution
\(\,\,\,\,\,\,e^{2x}+6e^x+8=0\)
\(\,\,\,\,\,\,\left(e^{x}+4\right)\left(e^{x}+2\right)=0\)
\(\,\,\,\,\,\,e^{x}+4=0\,\,\text{or}\,\,e^{x}+2=0\)
\(\,\,\,\,\,\,e^{x}=-4\,\,\text{or}\,\,e^{x}=-2\)
\(\,\,\,\,\,\,\ln{e^{x}}=\ln{-4}\,\,\text{or}\,\,\ln{e^{x}}=\ln{-2}\)
\(\,\,\,\,\,\,\)The answer is “No Solution”
\(\textbf{19)}\) \(4^{x+3} = 6^{x+2}\)
The answer is \(x = \displaystyle \frac{2\ln{6}-3\ln{4}}{\ln{4}-\ln{6}} \approx 5.4254\)
\(\,\,\,\,\,\,4^{x+3} = 6^{x+2}\)
\(\,\,\,\,\,\,\ln{4^{x+3}} = \ln{6^{x+2}}\)
\(\,\,\,\,\,\,(x+3)\ln{4} = (x+2)\ln{6}\)
\(\,\,\,\,\,\,x\ln{4}+3\ln{4} = x\ln{6}+2\ln{6}\)
\(\,\,\,\,\,\,x\ln{4}-x\ln{6} = 2\ln{6}-3\ln{4}\)
\(\,\,\,\,\,\,x\left(\ln{4}-\ln{6}\right) = 2\ln{6}-3\ln{4}\)
\(\,\,\,\,\,\,x = \displaystyle \frac{2\ln{6}-3\ln{4}}{\ln{4}-\ln{6}}\)
\(\,\,\,\,\,\,\)The answer is \(x \approx 1.41902\)
See Related Pages\(\)
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