Exam 3: Exponential and Logarithmic Functions

A sample contains 60 grams of carbon $\left( { } ^ { 14 } C \right) . { } ^ { 14 } C$ has a half-life of 5715 years. How much ${ } ^ { 14 } \mathrm { C }$ remains after 1900 years? Round your answer to three decimal places.

Find the exact value of $\ln e ^ { 1.50 } - \ln \sqrt { e }$ without using a calculator.

Use the properties of logarithms to expand the expression as a sum, difference, and/or constant multiple of logarithms. (Assume all variables are positive.) $\ln \sqrt [ 6 ] { t }$

Find the domain of the function below. $f ( x ) = \ln \left( \frac { x + 3 } { x - 5 } \right)$

Solve for $x : e ^ { x } \left( 8 - e ^ { x } \right) = 16$ . Round to 3 decimal places.

Determine whether or not $x = \frac { 1 } { 3 } \left( e ^ { - 3 } + 1 \right)$ is a solution to $\ln ( 3 x - 1 ) = - 3$ .

Match the function $y = 2 e ^ { - x }$ with its graph. Graph I: Graph II: Graph III: Graph IV: Graph V:

Solve the equation $f ( x ) = g ( x )$ algebraically. f(x)= g(x)=x-5

Evaluate the function $f ( x ) = \log _ { 2 } x$ at $x = \frac { 1 } { 2 }$ without using a calculator.

Solve the logarithmic equation below algebraically. Round your result to three decimal places. $1 + 2 \ln x = 6$

Solve the equation below algebraically. Round your result to three decimal places. $\frac { 3 + \ln x } { x ^ { 2 } } = 0$

Solve the exponential equation below algebraically. Round your result to three decimal places. $\frac { 500 } { 1 + e ^ { - x } } = 150$

The sales $S$ (in thousands of units) of a cleaning solution after $x$ hundred dollars is spent on advertising are given by $S = 20 \left( 1 - e ^ { k x } \right)$ . When $\$ 450$ is spent on advertising, 2500 units are sold. Complete the model by solving for $k$ and use the model to estimate the number of units that will be sold if advertising expenditures are raised to $\$ 650$ . Round your answer to the nearest unit.

Approximate the solution of $9 e ^ { x + 3 } = 18$ to 3 decimal places. (You may use a graphing utility.)

Find the exact value of $\log _ { 8 } \sqrt [ 3 ] { 64 }$ without using a calculator.

Match the function $y = \frac { 3 } { 1 + e ^ { - 2 x } }$ with its graph. Graph I: Graph II: Graph III: Graph IV: Graph V:

Solve the logarithmic equation below algebraically. $\ln ( x - 6 ) = \ln ( x + 6 ) - \ln ( x - 4 )$

Find the exact value of $\ln e ^ { 2.50 } - \ln \sqrt { e }$ without using a calculator.

Use the properties of logarithms to expand the expression as a sum, difference, and/or constant multiple of logarithms. (Assume all variables are positive.) $\ln \frac { x } { \sqrt [ 3 ] { x ^ { 5 } - 2 } }$

Determine whether the scatter plot below could best be modeled by a linear model, a quadratic model, an exponential model, a logarithmic model, or a logistic model.