Exam 5: Inverse, Exponential, and Logarithmic Functions

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 $\mathrm { pH } = 1.4$ .

The graph of a function f is given. Use the graph to find the indicated value. - $\mathrm { f } ^ { - 1 } ( 1 )$

Determine whether the statement is true or false. -If a function f has an inverse, then the graph of $f ^ { - 1 }$ can be obtained by reflecting the graph of f across the x-axis.

Solve the equation and express the solution in exact form. - $\log 5 x = \sqrt { \log 5 x }$

Solve the equation and express the solution in exact form. - $\ln ( 2 x - 1 ) + \ln ( x - 5 ) = \ln 5$

Decide whether the given functions are inverses. - f=\{(-3,-6),(0,4),(4,-3),(8,10)\} g=\{(-6,-3),(4,0),(-3,4),(10,8)\}

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

Determine whether or not the function is one-to-one. - $f ( x ) = x ^ { 2 } + 1$

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

In the formula N $\mathrm { N } = \mathrm { Ie }^ { kt }$ , N is the number of items in terms of an initial population I at a given time t and k is a growth constant equal to the percent of growth per unit time. How long will it take for the population of a Certain country to triple if its annual growth rate is 5.7%? Round to the nearest year.

Write the expression as a single logarithm with coefficient 1. Assume all variables represent positive real numbers with $a \neq 1 \text { and } b \neq 1$ . - $\frac { 1 } { 3 } \log _ { 3 } \left( x ^ { 6 } \right) + \frac { 1 } { 6 } \log _ { 3 } \left( x ^ { 6 } \right) - \frac { 1 } { 9 } \log _ { 3 } x$

Provide an appropriate response. -Give an equation of the form $f ( x ) = a ^ { x }$ to define the exponential function whose graph contains the point $\left( 4 , \frac { 1 } { 4096 } \right)$ . Assume that $a > 0$ .

Use the properties of logarithms to rewrite the expression. Simplify the result if possible. Assume all variables represent positive real numbers. - $\log _ { 12 } \left( \frac { 17 \sqrt { \mathrm { m } } } { \mathrm { n } } \right)$

Use a graphing calculator to estimate the solution set of the equation. Round to the nearest hundredth. - $\left( \frac { 1 } { 3 } \right) ^ { x } = 12$

Given $\log _ { 10 } 2 \approx 0.3010 \text { and } \log _ { 10 } 3 \approx 0.4771$ find the logarithm without using a calculator. - $\log 10 \sqrt { 24 }$

Provide an appropriate response. -Explain how the graph of $y = - 5 \left( \frac { 1 } { 4 } \right) ^ { x } \text { can be obtained from the graph of } y = 4 ^ { x }$

If the function is one-to-one, find its inverse. If not, write "not one-to-one." - $f ( x ) = \sqrt { x - 7 } , x \geq 7$

If $f ( x ) = 2 ^ { x }$ , find $f [ \log 2 ( \ln 2 ) ]$

Use a graphing calculator to estimate the solution set of the equation. Round to the nearest hundredth. - $3 ^ { x } = 15$