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Precalculus : Exponential and Logarithmic Functions

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Example Questions

Example Question #151 : Pre Calculus

Simplify the following expression so that it is only 1 logarithm:

3ln(4)+ln(3)-2ln(5)+4ln(1)

Possible Answers:

ln(1)

ln(4800)

$ln\left(\frac{45}{10}\right)$

$ln\left(\frac{192}{25}\right)$

ln(5)

Correct answer:

$ln\left(\frac{192}{25}\right)$

Explanation:

When combining logarithms there are a few rules to remember. First, addition outside a log is multiplication inside. Subtraction outside a log is division inside. Finally, multiplication/division outside are exponents inside. Also, ln(1)=0. So,

3ln(4)+ln(3)-2ln(5)+4ln(1)

turns into

ln(4^3)+ln(3)-ln(5^2)+0

and then

$ln\left(\frac{64\cdot 3}{25}\right)=ln\left(\frac{192}{25}\right)$

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Example Question #42 : Exponential And Logarithmic Functions

Suppose the graph of an exponential equation contains the points $\left ( 1, \frac{2}{3}\right )$ and $\left ( 2, \frac{4}{3}\right )$ . What is the formula of this line?

Possible Answers:

$y = 2\left ( \frac{1}{3} \right )^x$

$y = \frac{2}{3}x$

$y = \left ( \frac{2}{3} \right )^x$

$y = \frac{1}{3}(2)^x$

$y = \frac{3}{2}x - \frac{5}{6}$

Correct answer:

$y = \frac{1}{3}(2)^x$

Explanation:

Recall the standard form of an exponential equation:

y = ab^x

Plug in the two ordered pairs:

$\frac{2}{3} = ab^1$

$\frac{4}{3} = ab^2$

Solve for a in the first equation:

$\frac{2}{3}*\frac{1}{b} = ab^1 * \frac{1}{b}$

$\frac{2}{3b} = a$

Now plug this value into the second equation:

$\frac{4}{3} = \left ( \frac{2}{3b} \right )b^2$

Simplify:

$\frac{4}{3} = \frac{2}{3}b$

Multiply both sides by 3/2 and simplify:

2 = b

Plug this into the first equation to solve for a:

$\frac{2}{3}= a (2)^1$

$\frac{2}{3}*\frac{1}{2} = a$

$a = \frac{1}{3}$

Plug the values of a and b into the general form to get the answer:

$y = \frac{1}{3}(2)^x$

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Example Question #43 : Exponential And Logarithmic Functions

A calculator is required to solve this problem.

Suppose a population is currently at 5000, and that it increases by 5% every year. How large will the population be in 5 years? Round your answer to the nearest whole number.

Possible Answers:

6250

6381

5469

5250

5000

Correct answer:

6381

Explanation:

The standard formula for population growth is

$P(t) = P_{0}(1+r)^t$ ,

where P(t) is the population after time t, $P_{0}$ is the initial population, and r is the rate of growth as a decimal, per unit time. (In other words, r and t must have the same units.)

The problem also gives us initial population as 5000 and growth rate as 5% or .05.

Plug in:

P(5) = 5000(1+.05)^5

Simplify the parentheses:

P(5) = 5000(1.05)^5

Evaluate:

P(5) = 5000(1.27628...)

P(5) = 6381.4078....

Round to get the final answer:

P(5) = 6381

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Example Question #44 : Exponential And Logarithmic Functions

To solve this question you will need a calculator or other graphing tool capable of evaluating logarithms.

Suppose a colony of bacteria is decaying at a constant rate of 2% per minute. How many minutes will it take for the colony's population to decrease by half? Round your answer to the nearest whole minute.

Possible Answers:

20 minutes

35 minutes

-35 minutes

34 minutes

25 minutes

Correct answer:

34 minutes

Explanation:

We recall that the formula for population decay is

$P(t) = P_{0}(1-r)^t$ ,

where P(t) is the population at time t, $P_{0}$ is the initial population, and r is the rate of decrease per unit time (same unit as t).

P(t) is half of $P_{0}$ , so we can write

$P_{0}*\frac{1}{2} = P_{0}(1-.02)^t$ .

Simplify and eliminate common factors:

$\frac{1}{2}=.98^t$

Take the log of both sides. Note that the log has base .98.

$\log_{.98}\left ( \frac{1}{2}\right ) = t$

Use the change of base theorem to rewrite the log:

$\frac{\log_{10}\left ( \frac{1}{2}\right )}{\log_{10}.98} = t$

$t \approx 34.3096$

Round:

t = 34

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Example Question #45 : Exponential And Logarithmic Functions

Expand and simplify:

$\ln4e^{2x}$

Possible Answers:

$\ln4e^{2x}$

4+2x

$\ln4+2x$

$\ln4-2x$

Correct answer:

$\ln4+2x$

Explanation:

$\ln4e^{2x}\rightarrow \ln4+\ln(e^{2x})\rightarrow \ln4+2x$

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Example Question #155 : Pre Calculus

Identify the curve representing $f(x)=e^{3x-3}$ in the graph below.

2015-01-30_1313

Possible Answers:

Correct answer:

Explanation:

2015-01-30_1313

$f(x)=e^{3x-3}$ the -int will be when x=0 and the -int will not exist because all the values of this function are positive.

$f(0)=e^{-3}=.049$

So the curve must be D since that is the only curve that intersects at the point (0,.049)

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Example Question #1 : Rational Exponents

Simplify

$\frac{9^{\frac{2}{3}}\cdot 3^{\frac{5}{3}}}{6^{2}}$

Possible Answers:

$\frac{9}{2}$

$\frac{27}{2}$

$\frac{3}{4}$

$\frac{9}{4}$

Correct answer:

$\frac{3}{4}$

Explanation:

$\frac{9^{\frac{2}{3}}\cdot 3^{\frac{5}{3}}}{6^{2}}=\frac{(3^{2})^{\frac{2}{3}}\cdot3^{\frac{5}{3}}}{6^{2}}=\frac{3^{\frac{4}{3}}\cdot 3^{\frac{5}{3}}}{6^{2}}$

$=\frac{3^\frac{9}{3}}{6^2}=\frac{3^3}{6^2}=\frac{27}{36}=\frac{3}{4}$ .

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Example Question #1 : Simplify Expressions With Rational Exponents

Simplify the expression.

$\frac{x^{3/2}y^{7/3}}{x^3 y^{2/9}}$

Possible Answers:

$\frac{1}{x^{9/2}y^{14/27}}$

$x^{9/2}y^{23/9}$

$x^{3/2}y^{19/9}$

$\frac{y^{19/9}}{x^{3/2}}$

$\frac{x^{3/2}}{y^{19/9}}$

Correct answer:

$\frac{y^{19/9}}{x^{3/2}}$

Explanation:

Using the properties of exponents, we can either choose to subtract the exponents of the corresponding bases or rewrite the expression using negative exponents as such:

$x^{3/2}x^{-3}y^{7/3}y^{-2/9}$

Here, we combine the terms with corresponding bases by adding the exponents together to get

$x^{-3/2}y^{19/9}$

Placing the x term (since it has a negative exponent) in the denominator will result in the correct answer. It can be shown that simply subtracting the exponents of corresponding bases will result in the same answer.

$\frac{y^{19/9}}{x^{3/2}}$

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Example Question #2 : Simplify Expressions With Rational Exponents

Simplify the expression $4^{2.5}$ .

Possible Answers:

None of the other answers.

Correct answer:

Explanation:

We proceed as follows

$4^{2.5}$

Write 2.5 as a fraction

$4^{5/2}$

The denominator of the fraction is a , so it becomes a square root.

$(^2\sqrt{4})^5$

Take the square root.

2^5

Raise to the 5th power.

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Example Question #2 : Simplify Expressions With Rational Exponents

What is the value of $32^{\frac{3}{5}}$ ?

Possible Answers:

19.2

Correct answer:

Explanation:

Recall that when considering rational exponents, the denominator of the fraction tells us the "root" of the expression.

Thus in this case we are taking the fifth root of .

The fifth root of is , because $2\cdot2\cdot2\cdot2\cdot2 = 32$ .

Thus, we have reduced our expression to $2^3 = 2\cdot2\cdot2 = 8$ .

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Precalculus : Exponential and Logarithmic Functions

Study concepts, example questions & explanations for Precalculus

All Precalculus Resources

Example Questions

Example Question #151 : Pre Calculus

Example Question #42 : Exponential And Logarithmic Functions

Example Question #43 : Exponential And Logarithmic Functions

Example Question #44 : Exponential And Logarithmic Functions

Example Question #45 : Exponential And Logarithmic Functions

Example Question #155 : Pre Calculus

Example Question #1 : Rational Exponents

Example Question #1 : Simplify Expressions With Rational Exponents

Example Question #2 : Simplify Expressions With Rational Exponents

Example Question #2 : Simplify Expressions With Rational Exponents

All Precalculus Resources

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