Exam 5: Inverse, Exponential, and Logarithmic Functions

Use the definition of inverses to determine whether f and g are inverses. - $f ( x ) = 6 x ^ { 2 } + 1 , x \geq 0 ; \quad g ( x ) = \sqrt { \frac { x - 1 } { 6 } } , x \geq 1$

Find the function value. If the result is irrational, round your answer to the nearest thousandth. -Let $g ( x ) = \left( \frac { 1 } { 3 } \right) ^ { x }$ . Find $g ( 4.5 )$ .

Write an equivalent expression in exponential form. - $\log _ { ( x - 1 ) } 10 = 1$

Use the properties of logarithms to rewrite the expression. Simplify the result if possible. Assume all variables represent positive real numbers. - $\log _ { 6 } \left( \frac { \sqrt { 11 } } { 10 } \right)$

Find the present value of the future value. -Find the required annual interest rate, to the nearest tenth of a percent, for $1100 to grow to $1400 if interest is compounded monthly for 6 years.

Write an equivalent expression in exponential form. - $\log _ { 10 } 100 = 2$

Round your answer to the nearest tenth, when appropriate. Use the formula $\mathrm { pH } = - \log \left[ \mathrm { H } _ { 3 } \mathrm { O } ^ { + } \right] , \text {as }$ needed. -Find the $\mathrm { pH }$ if $\left[ \mathrm { H } _ { 3 } \mathrm { O } ^ { + } \right] = 3.5 \times 10 ^ { - 7 }$ .

Determine whether the statement is true or false. -If $\mathrm { f }$ is a one-to-one function and the graph of $\mathrm { f }$ lies completely within the third quadrant, then the graph of $\mathrm { f } ^ { - 1 }$ lies completely within the second quadrant.

Choose the one alternative that best completes the statement or answers the question. -How long will it take for prices in the economy to double at a $13 \%$ annual inflation rate?

Solve the equation and express the solution in exact form. - $\log _ { 9 } ( x - 8 ) + \log _ { 9 } ( x - 8 ) = 1$

Choose the one alternative that best completes the statement or answers the question. -A certain country's Consumer Price Index is approximated by $\mathrm { A } ( \mathrm { t } ) = 100 \mathrm { e } ^ { 0.024 \mathrm { t } }$ , where $\mathrm { t }$ represents the number of years. Use the function to determine the year in which costs will be $50 \%$ higher than in year 0.

Solve the equation and express the solution in exact form. - $\ln ( - x ) + \ln 4 = \ln ( 3 x - 9 )$

Use the properties of logarithms to rewrite the expression. Simplify the result if possible. Assume all variables represent positive real numbers. - $\log _ { 4 } \left( \frac { 4 \sqrt { 6 } } { 7 } \right)$

Use the definition of inverses to determine whether f and g are inverses. - $f ( x ) = \frac { 6 + x } { x } , \quad g ( x ) = \frac { 6 } { x - 1 }$

Graph the function. - $f(x)=\left(\frac{7}{2}\right)^{x}$

Find the function value. If the result is irrational, round your answer to the nearest thousandth. -Let $f ( x ) = \left( \frac { 1 } { 3 } \right) ^ { x }$ . Find $\left( \frac { 5 } { 2 } \right)$ .

Use a graphing calculator to estimate the solution set of the equation. Round to the nearest hundredth. - $\left( \frac { 1 } { 3 } \right) ^ { x } = 16$

Write an equation for the graph given. The graph represents an logarithmic function f with base 2 or 3, translated and/or reflected. -

Use properties of logarithms to evaluate the expression. - $\log _ { 10 } ( 0.0001 ) ^ { 5 }$

Solve the equation and express the solution in exact form. - $\log _ { 4 } ( x + 7 ) + \log _ { 4 } ( x - 7 ) = 1$