1.5.1: Exercises 1.5
- Page ID
- 62723
\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)
\( \newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\)
( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\)
\( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)
\( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\)
\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)
\( \newcommand{\Span}{\mathrm{span}}\)
\( \newcommand{\id}{\mathrm{id}}\)
\( \newcommand{\Span}{\mathrm{span}}\)
\( \newcommand{\kernel}{\mathrm{null}\,}\)
\( \newcommand{\range}{\mathrm{range}\,}\)
\( \newcommand{\RealPart}{\mathrm{Re}}\)
\( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)
\( \newcommand{\Argument}{\mathrm{Arg}}\)
\( \newcommand{\norm}[1]{\| #1 \|}\)
\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)
\( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\AA}{\unicode[.8,0]{x212B}}\)
\( \newcommand{\vectorA}[1]{\vec{#1}} % arrow\)
\( \newcommand{\vectorAt}[1]{\vec{\text{#1}}} % arrow\)
\( \newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vectorC}[1]{\textbf{#1}} \)
\( \newcommand{\vectorD}[1]{\overrightarrow{#1}} \)
\( \newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}} \)
\( \newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}} \)
\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)
\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Terms and Concepts
Exercise \(\PageIndex{1}\)
Explain the relationship between logarithmic functions and exponential functions.
- Answer
-
For the same base, they are inverses of each other.
Exercise \(\PageIndex{2}\)
What questions are you answering when you evaluate \(\log_5{(25)}\)?
- Answer
-
\(25\) is \(5\) raised to what power?
Exercise \(\PageIndex{3}\)
What is the value of the base for \(\ln{(x)}\)?
- Answer
-
\(e\)
Exercise \(\PageIndex{4}\)
Explain why logarithms help solve exponential statements.
- Answer
-
Logarithms help solve exponential statements because logarithms and exponentials are inverse functions.
Problems
Evaluate the given statement in exercises \(\PageIndex{5}\) – \(\PageIndex{8}\).
Exercise \(\PageIndex{5}\)
\(\displaystyle \log_3{(81)}\)
- Answer
-
\(4\)
Exercise \(\PageIndex{6}\)
\(\displaystyle \ln{(e^{5.7})}\)
- Answer
-
\(5.7\)
Exercise \(\PageIndex{7}\)
\(\displaystyle e^{-\ln{(x})}\)
- Answer
-
\(\displaystyle \frac{1}{x}\)
Exercise \(\PageIndex{8}\)
\(\displaystyle 4^{\log_2{(2^2)}}\)
- Answer
-
\(\displaystyle 16\)
Write the given statement as a single simplified logarithm in exercises \(\PageIndex{9}\) – \(\PageIndex{12}\).
Exercise \(\PageIndex{9}\)
\(\displaystyle 4 \log_3{(2x)}-\log_3{(y^2)}\)
- Answer
-
\(\displaystyle \log_3{\Big( \frac{16x^4}{y^2}\Big)}\)
Exercise \(\PageIndex{10}\)
\(\displaystyle \frac{2}{3} \ln{(x)} + 3\ln{(2y)}\)
- Answer
-
\(\displaystyle \ln{(8x^{2/3}y^3)}\)
Exercise \(\PageIndex{11}\)
\(\displaystyle (2x)\log_2{(3)}+ \log_2{(5)}\)
- Answer
-
\(\displaystyle \log_2{(5 (3^{2x}))}\)
Exercise \(\PageIndex{12}\)
\(\displaystyle 3\ln{(xy)}-2\ln{(x^2y)}\)
- Answer
-
\(\displaystyle \ln{\Big( \frac{y}{x} \Big)}\)
In exercises \(\PageIndex{13}\) – \(\PageIndex{17}\), solve the given problem for \(x\), if possible. If a problem cannot be solved, explain why.
Exercise \(\PageIndex{13}\)
\(\displaystyle 5^x = 25\)
- Answer
-
\(\displaystyle x=2\)
Exercise \(\PageIndex{14}\)
\(\displaystyle 5^x = -5\)
- Answer
-
Not possible; we cannot raise a positive number to a power and get a negative number.
Exercise \(\PageIndex{15}\)
\(\displaystyle 5^x = 0\)
- Answer
-
Not possible; we cannot raise a positive number to a power and get a zero.
Exercise \(\PageIndex{16}\)
\(\displaystyle 5^x = 0.2\)
- Answer
-
\(x=-1\)
Exercise \(\PageIndex{17}\)
\(\displaystyle 5^x = 1\)
- Answer
-
\(x=0\)
In exercises \(\PageIndex{18}\) – \(\PageIndex{25}\), solve the given problem for \(x\).
Exercise \(\PageIndex{18}\)
\(\displaystyle 3^{x-6} =2\)
- Answer
-
\(x=6+\log_3{(2)}\)
Exercise \(\PageIndex{19}\)
\(\displaystyle 4^{2x-5} = 3\)
- Answer
-
\(\displaystyle x=\frac{5+\log_4{(3)}}{2}\)
Exercise \(\PageIndex{20}\)
\(\displaystyle 2^{5x+6} = 4\)
- Answer
-
\(\displaystyle x=\frac{-4}{5}\)
Exercise \(\PageIndex{21}\)
\(\displaystyle 6^{x+\pi} = 2\)
- Answer
-
\(\displaystyle x=\log_6{(2)} - \pi\)
Exercise \(\PageIndex{22}\)
\(\displaystyle \bigg( \frac{1}{6}\bigg)^{-3x-2}=36^{x+1}\)
- Answer
-
\(\displaystyle x=0\)
Exercise \(\PageIndex{23}\)
\(\displaystyle -15=-8\ln{(3x)}+7\)
- Answer
-
\(\displaystyle x=\frac{1}{3} e^{11/4}\)
Exercise \(\PageIndex{24}\)
\(\displaystyle 2^x=3^{x-1}\)
- Answer
-
\(\displaystyle x=-\frac{\ln{(3)}}{\ln{(2)} -\ln{(3)}}\)
Exercise \(\PageIndex{25}\)
\(\displaystyle 8= 4\ln{(2x+5)}\)
- Answer
-
\(\displaystyle x=\frac{e^2-5}{2}\)