Exam 4: Inverse, Exponential, and Logarithmic Functions

Solve the problem. Round your answer to the nearest tenth, when appropriate. Use the formula $\mathrm { pH } = - \log \left[ \mathrm { H } _ { 3 } \mathrm { O } ^ { + } \right]$ as needed. -Find $\left[ \mathrm { H } _ { 3 } \mathrm { O } ^ { + } \right]$ if the $\mathrm { pH } = 5$ .

Write the word or phrase that best completes each statement or answers the question. -A lake is stocked with 559 fish of a new variety. The size of the lake, the availability of food, and the number of other fish restrict growth in the lake to a limiting value of 3493 . The population of fish in the lake after time $t$ , in months, is given by the function $P ( t ) = \frac { 3493 } { 1 + 4.27 e ^ { - 0.36 t } }$ . Find the population after 20 months.

Write in logarithmic form. - $\left( \frac { 11 } { 7 } \right) ^ { - 3 } = \frac { 343 } { 1331 }$

Solve for the indicated variable. - $\mathrm { i } = \mathrm { v } - \mathrm { k } \ln \mathrm { t }$ , for $\mathrm { t }$

Write the word or phrase that best completes each statement or answers the question. - $\text { Explain how the graph of } y = \log _ { 3 } \frac { x } { 2 } \text { can be obtained from the graph of } y = \log _ { 3 } x \text {. }$

Use the properties of logarithms to rewrite the expression. Simplify the result if possible. Assume all variables represent positive real numbers. - $\log _ { b } \sqrt [ 3 ] { \frac { x ^ { 8 } } { y ^ { 4 } z ^ { 7 } } }$

Write the word or phrase that best completes each statement or answers the question. -Suppose the number of Quickie hamburgers (in millions) served yearly can be modeled by $f ( x ) = 37.8 e ^ { 0.18 x }$ . Approximate how many years it takes for the number of hamburgers served to reach 77 million. Round to the nearest year.

Write the word or phrase that best completes each statement or answers the question. -A population is increasing according to the exponential function defined by $\mathrm { y } = 3 \mathrm { e }^{ .04 x }$ , where $\mathrm { y }$ is in millions and $\mathrm { x }$ is the number of years. Which of the following should be done in order to answer the question "When will the population reach 4 million?"

Write the expression as a single logarithm with coefficient 1. Assume all variables represent positive real numbers. - $\frac { 1 } { 2 } \log _ { 2 } \left( x ^ { 4 } \right) + \frac { 1 } { 4 } \log _ { 2 } \left( x ^ { 4 } \right) - \frac { 1 } { 6 } \log _ { 2 } x$

Write an equivalent expression in exponential form. - $\log _ { 243 } 3 = \frac { 1 } { 5 }$

Write the word or phrase that best completes each statement or answers the question. -A certain radioactive isotope has a half-life of approximately 1350 years. How many years to the nearest year would be required for a given amount of this isotope to decay to $55 \%$ of that amount?

The graph of a function f is given. Use the graph to find the indicated value. -

Solve the equation. Round to the nearest thousandth. - $4 ^ { 3 x } = 5 ( x + 1 )$

Match the function with its graph. - $f ( x ) = \log _ { 5 } \left( \frac { x } { 5 } \right)$

Solve the problem. -Which of the following is the same as $3 \log ( 4 x )$ for $x > 0$ ?

Find the value. Give an approximation to four decimal places. - $\log \left( \frac { 683 } { 363 } \right)$

Write the expression as a single logarithm with coefficient 1. Assume all variables represent positive real numbers. - $4 \log _ { 4 } ( 6 x - 2 ) + 6 \log _ { 4 } ( 2 x - 4 )$

Write in logarithmic form. - $\left( \frac { 7 } { 12 } \right) ^ { 5 } = \frac { 16,807 } { 248,832 }$

Evaluate the logarithm. - $\log _ { 10 } 100$

Find the function value. If the result is irrational, round your answer to the nearest thousandth. -Let $f ( x ) = 2 ^ { x }$ . Find $f ( 4.5 )$ .