Exam 3: Exponential and Logarithmic Functions

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.

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.

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

Find the domain of the function below. $f ( x ) = \sqrt { \ln ( x - 5 ) }$

Simplify the expression below. $6 + e ^ { \ln x ^ { 4 } }$

Solve $\ln x ^ { 2 } = 3$ for $x$ .

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.

Find the exponential model $y = a e ^ { b x }$ that fits the points shown in the table below. Round parameters to the nearest thousandth. x -2 0 y 48 3

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

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.

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

Determine whether or not $x = \frac { 6 } { 7 }$ is a solution to $3 ^ { 6 x - 3 } = 81$ .

Write the logarithmic equation below in exponential form. $\ln \sqrt [ 6 ] { e } = \frac { 1 } { 6 }$

Identify the vertical asymptote of the function $f ( x ) = 2 + \log ( x + 3 )$ .

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

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.

Solve the exponential equation below algebraically. $e ^ { - 6 x } = e ^ { x ^ { 2 } + 5 }$

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

Evaluate the function $f ( x ) = \log _ { 2 } x$ at $x = \frac { 1 } { 4 }$ without using a calculator.