Exam 5: Inverse, Exponential, and Logarithmic Functions

models salinity of ocean water to depths of 1000 meters at a certain latitude. x is the depth in meters and f(x) is in grams of salt per kilogram of seawater. Approximate the depth (to the Nearest tenth of a meter) where the salinity equals 35.

Determine whether or not the function is one-to-one. - $f ( x ) = \frac { 9 } { x - 1 }$

Use the properties of logarithms to rewrite the expression. Simplify the result if possible. Assume all variables represent positive real numbers. - $\log _ { 4 } ( 7 x + 11 y )$

Give the domain and range. - $f(x)=\log _{1 / 5}(x+4)$

Determine whether or not the function is one-to-one. -

If the function is one-to-one, find its inverse. If not, write "not one-to-one." - $\{ ( 3 , - 5 ) , ( 5 , - 3 ) , ( 9,2 ) , ( - 9 , - 2 ) \}$

Use the formula $\mathrm { D } = 10.0 \log ( \mathrm { S } / \mathrm { S } 0 )$ where the loudness of a sound in decibels is determined by S, the number of $\text { watt/m² }$ produced by the soundwave, and $\mathrm { S } _ { 0 } = 1.00 \times 10 ^ { - 12 } \mathrm { watt } / \mathrm { m } ^ { 2 }$ What is the intensity in $\text { watt/m² }$ of a noise measured at 40 decibels? (Round to the nearest tenth.)

Write an equivalent expression in exponential form. - $\log _ { 7 } \frac { 1 } { 49 } = x$

Use a graphing calculator to estimate the solution set of the equation. Round to the nearest hundredth. - $e ^ { x } = 8$

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

In year 0 , the number of female athletes participating in Summer Olympic-Type Games was 450 . In year 11 , about 3550 participated in the Summer Olympics in Atlanta. Assuming that P(0)=500 and that the exponential model applies, find the value of k rounded to the hundredths place, and write the function.

The number of reports of a certain virus has increased exponentially since year 0. The number of cases can be approximated using the function $\mathrm { r } ( \mathrm { t } ) = 142 \mathrm { e } ^ { 0.004 \mathrm { t } }$ , where t is the number of years since year 0. Estimate the Number of cases in year 40.

How long will it take for $800 to grow to $1300 at an interest rate of 5.6% if the interest is compounded quarterly? Round the number of years to the nearest hundredth.

Graph the exponential function using transformations where appropriate. -

If $f ( x ) = \ln x$ , find $f \left( e ^ { 16 \ln 9 } \right)$ .

Give the domain and range. - $f(x)=\log _{2} x$

Use the change of base rule to find the logarithm to four decimal places. - $\log _ { 3 } 0.681$

The growth in population of a city can be seen using the formula p(t) $\mathrm { p } ( \mathrm { t } ) = 2196 \mathrm { e } ^ { 0.005 \mathrm { t } }$ , where t is the number of years. According to this formula, how many years will it take the population to double its year 0 value? Round To the nearest tenth of a year.

Solve the following graphically. If necessary, round answers to the nearest thousandth. - $\log ( x + 1 ) - 3 = x ^ { 2 } - 6 x + 4$

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 the $\left[ \mathrm { H } _ { 3 } \mathrm { O } ^ { + } \right]$ if the $\mathrm { pH } = 2.3$ .