27.3: Review of exponential and logarithmic functions
- Page ID
- 68484
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)The population of a country grows exponentially at a rate of \(1\%\) per year. If the population was \(35.7\) million in the year \(2010\), then what is the population size of this country in the year \(2015\)?
- Answer
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\(37.5\) million
A radioactive substance decays exponentially at a rate of \(7\%\) per hour. How long do you have to wait until the substance has decayed to \(\dfrac 1 4\) of its original size?
- Answer
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\(19.1\) hours
Combine to an expression with only one logarithm.
- \(\dfrac{2}{3}\ln(x)+4\ln(y)\)
- \(\dfrac 1 2 \log_2(x)-\dfrac{3}{4}\log_2(y)+3\log_2(z)\)
- Answer
-
- \(\ln \left(\sqrt[3]{x^{2}} y^{4}\right)\)
- \(\log _{2}\left(\dfrac{\sqrt{x} z^{3}}{\sqrt[4]{y^{3}}}\right)\)
Assuming that \(x,y>0\), write the following expressions in terms of \(u=\log(x)\) and \(v=\log(y)\):
- \(\log\left(\dfrac{\sqrt[3]{x^4}}{y^2}\right)\)
- \(\log\left(x\sqrt{y^5}\right)\)
- \(\log\left(\sqrt[5]{xy^4}\right)\)
- Answer
-
- \(\dfrac{4}{3} u-2 v\)
- \(u+\dfrac{5}{2} v\)
- \(\dfrac{1}{5} u+\dfrac{4}{5} v\)
Solve without using the calculator: \(\log_3(x)+\log_3(x-8)=2\)
- Answer
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\(x=9\)
- Find the exact solution of the equation: \(6^{x+2}=7^x\)
- Use the calculator to approximate your solution from part (a).
- Answer
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- \(x=\dfrac{2 \log 6}{\log 7-\log 6}\)
- \(x \approx 23.25\)
\(45\)mg of fluorine-18 decay in \(3\) hours to \(14.4\)mg. Find the half-life of fluorine-18.
- Answer
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\(1.82\) hour
A bone has lost \(35\%\) of its carbon-\(14\). How old is the bone?
- Answer
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\(3561\) year
How much do you have to invest today at \(3\%\) compounded quarterly to obtain \(\$ 2,000\) in return in \(3\) years?
- Answer
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\(\$ 1,828.48\)
\(\$ 500\) is invested with continuous compounding. If \(\$692.01\) is returned after \(5\) years, what is the interest rate?
- Answer
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\(r=6.5 \%\)