27.3: Review of exponential and logarithmic functions
- Page ID
- 68484
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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
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- \(\ln \left(\sqrt[3]{x^{2}} y^{4}\right)\)
- \(\log _{2}\left(\dfrac{\sqrt{x} z^{3}}{\sqrt[4]{y^{3}}}\right)\)
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
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- \(\dfrac{4}{3} u-2 v\)
- \(u+\dfrac{5}{2} v\)
- \(\dfrac{1}{5} u+\dfrac{4}{5} v\)
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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\(r=6.5 \%\)