Exam 14: Exponential and Logarithmic Functions

An equation of the form $f ( x ) = b ^ { x }$ is a one-to-one function.

Express $\log b \left( \frac { x ^ { \frac { 1 } { 3 } } y ^ { \frac { 1 } { 6 } } } { z ^ { 9 } } \right)$ as the sum or difference of simpler logarithmic quantities. Assume that all variables represent positive real numbers.

The change-of-base formula states that if a , b , and r are positive numbers with $a \neq 1$ and $b \neq 1$ , then $\log _ { a } r = \frac { \log _ { b } a } { \log _ { b } r }$

True or false? The graph below represents a one-to-one function.

Solve the equation. $4 ^ { 2 x } = 16$

Evaluate the expression. $\log _ { 40,353,607 } \sqrt { 40,353,607 }$

Use your calculator to find x when given ln x . Please give the answer to four decimal places. $\ln x = - 2.7752$

Solve the equation. $10 ^ { x } = 0.000001$

For a certain strain of bacteria, the number of bacteria present after t hours is given by the equation Q = Q _oe^{0.44 t} where Q _o represents the initial number of bacteria. How long will it take 400 bacteria to increase to 4,000 bacteria? Express answer to the nearest tenth without the units. __________ hours

Solve the equation. $\left( 81 ^ { x } \right) \left( 27 ^ { 12 x - 1 } \right) = 2187$ Please enter your answer as a fraction.

Solve the equation. $6 ^ { 2 x - 1 } = \frac { 1 } { 6 }$

Form the inverse function f ^{- -1}, and list the domain and range of f ^{- -1}. f = {(1, 6), (4, 8), (6, 50)} Match the name of each set with the corresponding set. - $\{ 6,8,50 \}$

Use a calculator to find each common logarithm. Express answer to four decimal places. $\log 3420.7$

Use the formulas $A = p \left( 1 + \frac { r } { n } \right) ^ { n t }$ or $A = p \mathrm { e } ^ { r t }$ to find the amount for the investment. $4,200 for 5 years at 5% compounded continuously.

Evaluate the expression. $\log _ { 5 } \sqrt { 3,125 }$

Find f - -1 ( x ). $f ( x ) = \frac { 2 } { 3 } x - \frac { 9 } { 10 }$

Solve the equation. log 28 ( x + 1 ) + log 28 ( x $-$ 32 ) = 2

Form the inverse function f ^{- -1}, and list the domain and range of f ^{- -1}. f = {(0, 0), (2, 6), (-1, -1), (-2, -6)} Match the name of each set in the left column with the corresponding set in the right column. - $\{ 0,2 , - 1 , - 2 \}$

Use the formula $A = p \left( 1 + \frac { r } { n } \right) ^ { n t }$ to find the amount for the investments. $13,600 for 5 years at 4.5% compounded semiannually. Please round the answer to the nearest cent. $A =$ $__________ $13,600 for 5 years at 4% compounded monthly. Please round the answer to the nearest cent. $A =$ $__________ Determine which investment amounts to more. __________

For a certain culture the equation $Q ( t ) = Q _ { 0 } e ^ { 0.4 t }$ , where Q 0 is an initial number of bacteria and t is time measured in hours, yields the number of bacteria as a function of time. How long will it take 700 bacteria to increase to 2,800? Please enter the answer as a number without the units and round to the nearest tenth. __________ hours