Exam 4: Exponential and Logarithmic Functions

Solve the problem. -The $\mathrm { pH }$ of a solution ranges from 0 to 14 . An acid has a pH less than 7. Pure water is neutral and has a pH of 7 . The $\mathrm { pH }$ of a solution is given by $\mathrm { pH } = - \log \mathrm { x }$ where $x$ represents the concentration of the hydrogen ions in the solution in moles per liter. Find the hydrogen ion concentration if the $\mathrm { pH } = 2.4$ .

Use properties of logarithms to condense the logarithmic expression. Write the expression as a single logarithm whose coefficient is 1. Where possible, evaluate logarithmic expressions. - $\log _ { \mathrm { c } } \mathrm { t } + \log _ { \mathrm { c } } \mathrm { s }$

The graph of an exponential function is given. Select the function for the graph from the functions listed. -

Use properties of logarithms to condense the logarithmic expression. Write the expression as a single logarithm whose coefficient is 1. Where possible, evaluate logarithmic expressions. - $\left( \log _ { a } m - \log _ { a } n \right) + 4 \log _ { a } k$

Use properties of logarithms to expand the logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. - $\log _ { a } \left( \frac { x ^ { 4 } \sqrt [ 3 ] { x + 5 } } { ( x - 2 ) ^ { 2 } } \right)$

Use the Quotient Rule Use properties of logarithms to expand the logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. - $\log _ { 3 } \left( \frac { 2 } { 5 } \right)$

Solve the equation by expressing each side as a power of the same base and then equating exponents. - $4 ^ { ( 7 - 3 x ) } = \frac { 1 } { 16 }$

Evaluate or simplify the expression without using a calculator. - $2 \log 10 ^ { 3.9 }$

Use properties of logarithms to expand the logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. - $\ln \sqrt [ 4 ] { \mathrm { ey } }$

Use common logarithms or natural logarithms and a calculator to evaluate to four decimal places - $\log _ { 28 } 321$

Solve the problem. -The $\mathrm { pH }$ of a solution ranges from 0 to 14 . An acid has a pH less than 7. Pure water is neutral and has a pH of 7 . The $\mathrm { pH }$ of a solution is given by $\mathrm { pH } = - \log \mathrm { x }$ where $\mathrm { x }$ represents the concentration of the hydrogen ions in the solution in moles per liter. Find the $\mathrm { pH }$ if the hydrogen ion concentration is $9.5 \times 10 - 10$ .

Use the Power Rule Use properties of logarithms to expand the logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. - $\log _ { \mathrm { d } } \mathrm { x } ^ { 10 }$

Graph the function. - $\text { Use the graph of } f ( x ) = e ^ { x } \text { to obtain the graph of } g ( x ) = e ^ { x / 3 } - 1$

Solve the exponential equation. Use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. - $3 \mathrm { e } ^ { \mathrm { x } } = 29$

Approximate the number using a calculator. Round your answer to three decimal places. - $2 ^ { - 1.3 }$

Use the Definition of a Logarithm to Solve Logarithmic Equations Solve the logarithmic equation. Be sure to reject any value that is not in the domain of the original logarithmic expressions. Give the exact answer. - $2 + 2 \ln x = 7$

Solve the equation by expressing each side as a power of the same base and then equating exponents. - $9 ^ { x + 10 } = 27 ^ { x - 4 }$

Graph the function. - $\text { Use the graph of } f ( x ) = e ^ { x } \text { to obtain the graph of } g ( x ) = e ^ { x + 4 } - 4 \text {. }$

Model Exponential Growth and Decay Solve. -The function $\mathrm { A } = \mathrm { A } _ { 0 } \mathrm { e } ^ { - 0.0077 \mathrm { x } }$ models the amount in pounds of a particular radioactive material stored in a concrete vault, where $x$ is the number of years since the material was put into the vault. If 300 pounds of the material are initially put into the vault, how many pounds will be left after 140 years?

Solve the exponential equation. Use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. - $\mathrm { e } ^ { 3 \mathrm { x } } = 5$