Logarithmic and Exponential Equations – Questions with Solutions
Below is a collection of practice questions involving exponential and logarithmic expressions and equations.
All solutions are provided at the bottom of the page and are solved without using a calculator.
Question 1 – Exponential Expressions
Evaluate the following expressions without using a calculator:
A. \(\;10^{4^{0.5}}\)
B. \(\;10^{\frac{\log(9)}{2}}\)
C. \(\;e^{3\ln(4)}\)
D. \(\;10^{-2\log(4)}\)
E. \(\;e^{2\ln(4)-\ln(16)}\)
Question 2 – Logarithmic Expressions
Evaluate the following logarithmic expressions:
A. \(\;\log_{9}(3)\)
B. \(\;\log_{2}\!\left(\frac{1}{8}\right)\)
C. \(\;\log_{2}(400)-\log_{2}(100)\)
D. \(\;\log_{8}(4)\)
E. \(\;\log_{5}(600)-\log_{5}(24)\)
Question 3 – Exponential Equations
Solve the following
exponential equations
:
A. \(\;e^{x}=4\)
B. \(\;e^{x}=\pi^{2}\)
C. \(\;4^{\log_{4}(3x-2)}=10\)
D. \(\;10^{\log_{10}(3x)}=15\)
E. \(\;3^{x+1}+3^{x}+3^{x-1}=39\)
Question 4 – Logarithmic Equations
Solve the following
logarithmic equations
:
A. \(\;\ln(\ln x)=4\)
B. \(\;\ln(x)-\ln(4)=2\ln(x)-\ln(16)\)
C. \(\;(\ln x)^5=25^{\frac{5}{2}}\)
D. \(\;\log(x-1)=\log(6)-\log(x)\)
E. \(\;\log_{3}(3^{x+1}-18)=2\)
F. \(\;\log_{7}(x)+\log_{x}(7)=2\)
G. \(\;\log_{x}(27)=\frac{3}{4}\)
Solutions
Solution to Question 1
\[
\begin{aligned}
\text{A. } &10^{4^{0.5}} = 10^{4 \times 0.5} = 10^2 = 100 \\[6pt]
\text{B. } &10^{\frac{\log(9)}{2}} = 10^{\log(9^{1/2})} = 3 \\[6pt]
\text{C. } &e^{3\ln(4)} = e^{\ln(4^3)} = 4^3 = 64 \\[6pt]
\text{D. } &10^{-2\log(4)} = 10^{\log(4^{-2})} = 4^{-2} = \frac{1}{16} \\[6pt]
\text{E. } &e^{2\ln(4)-\ln(16)} = e^{\ln(16)-\ln(16)} = e^0 = 1
\end{aligned}
\]
Solution to Question 2
\[
\begin{aligned}
\text{A. } &\log_{9}(3)=\log_{9}(9^{1/2})=\frac{1}{2} \\[6pt]
\text{B. } &\log_{2}\!\left(\frac{1}{8}\right)=\log_{2}(2^{-3})=-3 \\[6pt]
\text{C. } &\log_{2}(400)-\log_{2}(100)=\log_{2}(4)=2 \\[6pt]
\text{D. } &\log_{8}(4)=\log_{8}(8^{2/3})=\frac{2}{3} \\[6pt]
\text{E. } &\log_{5}(600)-\log_{5}(24)=\log_{5}(25)=2
\end{aligned}
\]
Solution to Question 3
\[
\begin{aligned}
\text{A. } &x=\ln(4) \\[6pt]
\text{B. } &x=\ln(\pi^2)=2\ln(\pi) \\[6pt]
\text{C. } &3x-2=10 \Rightarrow x=4 \\[6pt]
\text{D. } &3x=15 \Rightarrow x=5 \\[6pt]
\text{E. } &3^{x+1}+3^{x}+3^{x-1}=39 \\
&\Rightarrow 3^{x}(3+1+\tfrac{1}{3})=39 \\
&\Rightarrow 3^{x}=9 \Rightarrow x=2
\end{aligned}
\]
Solution to Question 4
\[
\begin{aligned}
\text{A. } &\ln x = e^4 \Rightarrow x=e^{e^4} \\[6pt]
\text{B. } &\ln\!\left(\frac{x}{4}\right)=\ln\!\left(\frac{x^2}{16}\right)
\Rightarrow x=4 \\[6pt]
\text{C. } &(\ln x)^5=5^5 \Rightarrow \ln x=5 \Rightarrow x=e^5 \\[6pt]
\text{D. } &\log(x-1)=\log\!\left(\frac{6}{x}\right)
\Rightarrow x=3 \\[6pt]
\text{E. } &3^{x+1}-18=9 \Rightarrow x=2 \\[6pt]
\text{F. } &\frac{\ln 7}{\ln x}+\frac{\ln x}{\ln 7}=2
\Rightarrow (\ln 7-\ln x)^2=0 \Rightarrow x=7 \\[6pt]
\text{G. } &x^{3/4}=27 \Rightarrow x=3^4
\end{aligned}
\]
More Practice and References
More math questions with detailed solutions