# 5.4.1: Logarithms and Logarithmic Functions (Exercises)

$$\newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$$ $$\newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$$$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}[1]{\| #1 \|}$$ $$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}[1]{\| #1 \|}$$ $$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$$$\newcommand{\AA}{\unicode[.8,0]{x212B}}$$

## SECTION 5.4 PROBLEM SET: LOGARITHMS AND LOGARITHMIC FUNCTIONS

Rewrite each of these exponential expressions in logarithmic form:

 $$3^{4}=81$$ $$10^{5}=100,000$$ $$5^{-2}=0.04$$ $$4^{-1}=0.25$$ $$16^{1 / 4}=2$$ $$9^{1 / 2}=3$$

Rewrite each of these logarithmic expressions in exponential form:

 $$\log _{5} 625=4$$ $$\log _{2}(1 / 32)=-5$$ $$\log _{11} 1331=3$$ $$\log _{10} 0.0001=-4$$ $$\log _{64} 4=1 / 3$$ $$\ln \sqrt{e}=\frac{1}{2}$$

If the expression is in exponential form, rewrite it in logarithmic form.

If the expression is in logarithmic form, rewrite it in exponential form.

 $$5^{x}=15625$$ $$x=9^{3}$$ $$\log _{5} 125=x$$ $$\log _{3} x=5$$ $$\log _{10} y=4$$ $$\mathrm{e}^{\mathrm{x}}=10$$ $$\ln x=-1$$ $$e^{5}=y$$

For each equation, rewrite in exponential form and solve for x.

 $$\log _{5}(x)=3$$ $$\log _{2}(x)=-2$$ $$\log _{10}(x)=-3$$ $$\log _{3}(x)=6$$ $$\log _{25}(x)=1 / 2$$ $$\log _{64}(x)=1 / 3$$

 $$\ln \sqrt[3]{e}$$ $$\ln \frac{1}{e^{2}}$$ $$\ln \mathrm{e}^{10}$$ $$\log _{10}\left(10^{e}\right)$$
 $$\log 20$$ $$\ln 42$$ $$\ln 2.9$$ $$\log 0.5$$ $$\log _{4} 36$$ $$\log _{7} 100$$ $$\log _{105} 3.5$$ $$\log _{1.067} 2$$