# 27.3: Review of exponential and logarithmic functions

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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 $$2015$$?

$$37.5$$ 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?

$$19.1$$ hours

## Exercise $$\PageIndex{3}$$

Combine to an expression with only one logarithm.

1. $$\dfrac{2}{3}\ln(x)+4\ln(y)$$
2. $$\dfrac 1 2 \log_2(x)-\dfrac{3}{4}\log_2(y)+3\log_2(z)$$
1. $$\ln \left(\sqrt{x^{2}} y^{4}\right)$$
2. $$\log _{2}\left(\dfrac{\sqrt{x} z^{3}}{\sqrt{y^{3}}}\right)$$

## Exercise $$\PageIndex{4}$$

Assuming that $$x,y>0$$, write the following expressions in terms of $$u=\log(x)$$ and $$v=\log(y)$$:

1. $$\log\left(\dfrac{\sqrt{x^4}}{y^2}\right)$$
2. $$\log\left(x\sqrt{y^5}\right)$$
3. $$\log\left(\sqrt{xy^4}\right)$$
1. $$\dfrac{4}{3} u-2 v$$
2. $$u+\dfrac{5}{2} v$$
3. $$\dfrac{1}{5} u+\dfrac{4}{5} v$$

## Exercise $$\PageIndex{5}$$

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

$$x=9$$

## Exercise $$\PageIndex{6}$$

1. Find the exact solution of the equation: $$6^{x+2}=7^x$$
2. Use the calculator to approximate your solution from part (a).
1. $$x=\dfrac{2 \log 6}{\log 7-\log 6}$$
2. $$x \approx 23.25$$

## Exercise $$\PageIndex{7}$$

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

$$1.82$$ hour

## Exercise $$\PageIndex{8}$$

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

$$3561$$ 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?

$$\ 1,828.48$$

## Exercise $$\PageIndex{10}$$

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

$$r=6.5 \%$$