Exam 3: Exponential and Logarithmic Functions

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

Simplify the expression below. $1 + e ^ { \ln x ^ { 6 } }$

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

Determine whether or not $x = \frac { 4 } { 7 }$ is a solution to $3 ^ { 4 x - 3 } = 81$ .

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

A sample contains 75 grams of carbon $\left( { } ^ { 14 } C \right) . { } ^ { 14 } C$ has a half-life of 5715 years. How much ${ } ^ { 14 } \mathrm { C }$ remains after 100 years? Round your answer to three decimal places.

Identify the graph that represents the function. $y = 5 e ^ { - 2 ( x - 1 ) ^ { 2 } }$

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

What is the value of the function $f ( x ) = 3.8 e ^ { - 1.5 x }$ at $x = 2.5$ ? Round to 3 decimal places.

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

Solve the equation below algebraically. Round your result to three decimal places. $e ^ { x } + 4 x e ^ { x } = 0$

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

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

Solve $\ln x ^ { 2 } - \ln 3 = 0$ 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 domain of the function below. $f ( x ) = \ln \left( \frac { x - 4 } { x + 5 } \right)$

Solve the equation below algebraically. $- x ^ { 2 } e ^ { 2 x } - 7 x e ^ { 2 x } = 0$

Identify the graph of the function. $f ( x ) = \left( \frac { 1 } { 2 } \right) ^ { x }$

Determine whether or not $x = \frac { 4 } { 7 }$ is a solution to $3 ^ { 4 x - 3 } = 81$ .

Solve $\left( \frac { 1 } { 4 } \right) ^ { x } = 64$ for $x$ .