Exam 4: Exponential and Logarithmic Functions

ng, in -The population of a country in 2010 was approximately 35 million with an annual growth rate of $0.804 \%$ . At this rate, the population $P ( t )$ (in millions) can be approximated by $P ( t ) = 35 ( 1.00809 ) ^ { t }$ where $t$ is the time in years since 2010 . a. Evaluate $P ( 0 )$ and interpret its meaning in the context of this problem. b. Evaluate $P ( 6 )$ and interpret its meaning in the context of this problem. Round the population value to the nearest million.

Write the logarithm as a sum or difference of logarithms. Simplify each term as much as possible. - $\log _ { 6 } x ^ { 4 } y ^ { 5 }$

A one-to-one function is given. Write an expression for the inverse function. - $f ( x ) = \frac { 6 - x } { 2 }$

Approximate the value of the logarithm to four decimal places. - $\log 0.00052$

Write the word or phrase that best completes each statement or answers the question. Provide the missing information. -An equation such as $4 ^ { x } = 9$ is called an equation because the equation contains a variable in the exponent.

Write the word or phrase that best completes each statement or answers the question. Provide the missing information. -The horizontal asymptote of an exponential function $f ( x ) = b ^ { x }$ is the line .

Solve the problem. -Use the graph of $y = 3 ^ { x }$ to graph the function. Write the domain and range in interval notation. $f ( x ) = 3 ^ { x + 5 } + 2$

Write the word or phrase that best completes each statement or answers the question. Provide the missing information. -The logarithmic function base 10 is called the ــــــــــــــlogarithmic function, and the logarithmic function base $e$ is called the ـــــــــــــــlogarithmic function.

Solve the equation. Write the solution set with the exact solutions. Also give approximate solutions to 4 decimal places if necessary. - $3 ^ { | x | } - 6 = 3$

Solve the logarithmic equation. - $19 = 17 - \log _ { 2 } ( 2 x + 4 )$

Approximate $f ( x ) = \ln x$ for the given value of x. Round to four decimal places. - $f ( 9 \pi )$

Simplify the expression. - $\ln \left( \frac { 1 } { e ^ { 2 } } \right)$

ng, in -A radioactive isotope $\mathrm { X }$ is a by-product of nuclear fission with a half-life of $28.7 \mathrm { yr }$ . After a nuclear accident, large areas surrounding the site were contaminated with isotope X. If 10 micrograms of $X$ is present in a sample, the function $A ( t ) = 10 \left( \frac { 1 } { 2 } \right) ^ { t / 28.7 }$ gives the amount $A ( t )$ , in micrograms, present after $t$ years. Evaluate the function for the given values of $t$ and interpret the meaning in context. a. $A ( 57.4 ) \quad$ b. $A ( 100 )$

The function $f : = \{ ( 1,5 ) , ( - 2,3 ) , ( - 4,2 ) , ( 2,5 ) \} \text { (is/is not) a one-to-one function. }$

Use the definition of a one-to-one function to determine if the function is one-to-one. - $f ( x ) = - 2 x ^ { 2 } - 13$

Write the domain in interval notation. - $f ( x ) = \log _ { 9 } \left( \frac { x - 5 } { x + 8 } \right)$

Write the logarithmic expression as a single logarithm with coefficient 1, and simplify as much as possible. - $\frac { 1 } { 3 } \log _ { 7 } a + \log _ { 7 } \left( b ^ { 2 } - 25 \right) - \log _ { 7 } ( b + 5 )$

Solve the equation. Write the solution set with the exact solutions. Also give approximate solutions to 4 decimal places if necessary. - $3 ^ { x ^ { 2 } - 7 } = 9$

Solve the equation. - $\frac { 10 ^ { x } - 31 \cdot 10 ^ { - x } } { 5 } = 6$

Choose the one alternative that best completes the statement or answers the question. Solve the equation. - $2 ^ { 2 x } = 4 ^ { x ^ { 2 } }$