27.3: Review of exponential and logarithmic functions

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Thomas Tradler and Holly Carley
CUNY New York City College of Technology via New York City College of Technology at CUNY Academic Works

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Exercise $\PageIndex{1}$

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: $37.5$ million

Exercise $\PageIndex{2}$

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: $19.1$ hours

Exercise $\PageIndex{3}$

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

$\ln \left(\sqrt[3]{x^{2}} y^{4}\right)$
$\log _{2}\left(\dfrac{\sqrt{x} z^{3}}{\sqrt[4]{y^{3}}}\right)$

Exercise $\PageIndex{4}$

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

$\dfrac{4}{3} u-2 v$
$u+\dfrac{5}{2} v$
$\dfrac{1}{5} u+\dfrac{4}{5} v$

Exercise $\PageIndex{5}$

Solve without using the calculator: $\log_3(x)+\log_3(x-8)=2$

Answer: $x=9$

Exercise $\PageIndex{6}$

Find the exact solution of the equation: $6^{x+2}=7^x$
Use the calculator to approximate your solution from part (a).

Answer

$x=\dfrac{2 \log 6}{\log 7-\log 6}$
$x \approx 23.25$

Exercise $\PageIndex{7}$

$45$mg of fluorine-18 decay in $3$ hours to $14.4$mg. Find the half-life of fluorine-18.

Answer: $1.82$ hour

Exercise $\PageIndex{8}$

A bone has lost $35\%$ of its carbon-$14$. How old is the bone?

Answer: $3561$ year

Exercise $\PageIndex{9}$

How much do you have to invest today at $3\%$ compounded quarterly to obtain $\$ 2,000$ in return in $3$ years?

Answer: $\$ 1,828.48$

Exercise $\PageIndex{10}$

$\$ 500$ is invested with continuous compounding. If $\$692.01$ is returned after $5$ years, what is the interest rate?

Answer: $r=6.5 \%$

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