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.

Find the exact value of the logarithm without using a calculator, if possible. $\ln \frac { 1 } { \sqrt [ 2 ] { e } }$

Match the function $y = 1 + \ln ( x + 2 )$ with its graph. Graph I: Graph II: Graph III: Graph IV: Graph V:

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

Identify the value of the function $f ( x ) = \log _ { 10 } x$ at $x = 715$ . Round to 3 decimal places.

Solve the equation below algebraically. Round your result to three decimal places. $\frac { 1 + 2 \ln x } { x ^ { 5 } } = 0$

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

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

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 or not $x = \frac { 1 } { 3 } \left( e ^ { - 2 } + 1 \right)$ is a solution to $\ln ( 3 x - 1 ) = - 2$ .

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

Condense the expression below to the logarithm of a single quantity. $4 [ \ln x - \ln ( x + 4 ) - \ln ( x - 4 ) ]$

Find the domain of the function below. $f ( x ) = \sqrt { \ln ( x + 2 ) }$

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

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

Find the domain of the function below. $f ( x ) = \ln \left( \frac { x } { x ^ { 2 } + 16 } \right)$

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

The atmospheric pressure decreases with increasing altitude. At sea level, the average air pressure is approximately $1.03323$ kilograms per square centimeter, and this pressure is called one atmosphere. Variations in weather conditions cause changes in the atmospheric pressure of up to $\pm 5$ percent. The table below shows the pressures $p$ (in atmospheres) for various altitudes $h$ (in kilometers). Use the regression feature of a graphing utility to find the logarithmic model $h = a + b \ln p$ for the data. Use the model to estimate the altitude at which the pressure is $0.28$ atmosphere. Round your answer to two decimal places.

Use the regression feature of a graphing utility to find a logarithmic model $y = a + b \ln x$ for the data. $( 1,3 ) , ( 2,4 ) , ( 3,4.5 ) , ( 4,5 ) , ( 5,5.1 ) , ( 6,5.2 ) , ( 7,5.5 )$

Solve the exponential equation below algebraically. Round your result to three decimal places. $\frac { 375 } { 1 + e ^ { x } } = 125$