Exam 4: Exponential and Logarithmic Functions

Choose the one alternative that best completes the statement or answers the question. - $\mathrm { pH } = - \log _ { 10 } \left[ \mathrm { H } ^ { + } \right]$ Find the $\mathrm { pH }$ if the $\left[ \mathrm { H } ^ { + } \right] = 3.9 \times 10 ^ { - 8 }$ .

Evaluate the expression using the values given in the table. - $( \mathrm { g \circ f ) }( 1 )$ x 1 6 10 12 f(x) -2 10 3 12 -5 -2 1 3 () 1 -5 6 10

Express as a single logarithm. - $8 \log _ { \mathrm { b } } 8 + 9 \log _ { \mathrm { b } } 3$

Choose the one alternative that best completes the statement or answers the question. -The value of a particular investment follows a pattern of exponential growth. In the year 2000, you invested money in a money market account. The value of your investment t years after 2000 is given by the exponential ) By what percentage is the account increasing each year? $\text { growth model } \mathrm { A } = 8,500 \mathrm { e } ^ { 0.065 \mathrm { t } }$

Find the value of the expression. -Let $\log _ { \mathrm { b } } \mathrm { A } = 3.455$ and $\log _ { \mathrm { b } } \mathrm { B } = 0.192$ . Find $\log _ { \mathrm { b } } \frac { \mathrm { A } } { \mathrm { B } }$ .

For the given functions f and g, find the requested composite function. - $f ( x ) = \frac { 5 } { x - 1 } , \quad g ( x ) = \frac { 1 } { 6 x } ; \quad$ Find $( f \circ g ) ( x )$

Write as the sum and/or difference of logarithms. Express powers as factors. - $\log _ { 4 } \sqrt { 7 x }$

Write the word or phrase that best completes each statement or answers the question. -Between 8:30 a.m. and 9:30 a.m., cars drive through the Cappuccino Express at a rate of 12 cars per hour (0.2 per minute). The following formula from probability can be used to determine the probability that a car will arrive within t minutes of 8:30a.m. $\mathrm { F } ( \mathrm { t } ) = 1 - \mathrm { e } ^ { - 0.2 \mathrm { t } }$ Determine how many minutes are needed for the probability to reach 0.6.

Determine i) the domain of the function, ii) the range of the function, iii) the domain of the inverse, and iv) the range of the inverse. - $f ( x ) = \sqrt { 5 x + 4 }$

Choose the one alternative that best completes the statement or answers the question. -During 1991, 200,000 people visited Rave Amusement Park. During 1997 , the number had grown to 834,000 . If the number of visitors to the park obeys the law of uninhibited growth, find the exponential growth function that models this data.

Choose the one alternative that best completes the statement or answers the question. Graph the function using a graphing utility and the Change-of-Base Formula. - $y=\log _{3}(x-2)$

Determine i) the domain of the function, ii) the range of the function, iii) the domain of the inverse, and iv) the range of the inverse. - $f ( x ) = - 7 x - 4$

Choose the one alternative that best completes the statement or answers the question. Indicate whether the function is one-to-one. -{(12, -18), (-12, -18), (20, -8)}

Write the word or phrase that best completes each statement or answers the question. -A biologist has a bacteria sample. She records the amount of bacteria every week for 8 weeks and finds that the exponential function of best fit to the data is $\mathrm { A } = 150 \cdot 1.79 ^ { \mathrm { t } }$ . Express the function of best fit in the form $A = A _ { 0 } e ^ { k t } \text {. }$

Solve the equation. - $\frac { 1 } { 3 } \log _ { 2 } ( x + 6 ) = \log _ { 8 } ( 3 x )$

Find the inverse of the function and state its domain and range . - $\{ ( - 3,4 ) , ( - 1,5 ) , ( 0,2 ) , ( 2,6 ) , ( 5,7 ) \}$

For the given functions f and g, find the requested composite function value. - $f ( x ) = 2 x + 2 , \quad g ( x ) = 2 x ^ { 2 } + 1 ; \quad \text { Find } ( f \circ f ) ( 0 )$

Choose the one alternative that best completes the statement or answers the question. Solve the exponential equation. Use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. - $3 ^ { ( 2 x - 1 ) } = 19$

Graph the function. - $f ( x ) = \frac { 1 } { 5 } \cdot 5 ^ { x }$

Write the word or phrase that best completes each statement or answers the question. -The volume $\mathrm { V }$ (in cubic inches) of a cylindrical pipe with length 12 inches is given by $\mathrm { V } ( \mathrm { r } ) = 12 \pi \mathrm { r } ^ { 2 }$ , where $\mathrm { r }$ is the radius of the piston (in inches). If the radius is increasing with time $t$ (in minutes) according to the formula $r ( t ) = \frac { 1 } { 6 } t ^ { 2 } , t \geq 0$ , find the volume $V$ of the pipe as a function of the time $t$ .