Exam 4: Exponential and Logarithmic Functions

For the given functions f and g, find the requested composite function value. - $f ( x ) = \sqrt { x + 2 } , \quad g ( x ) = 3 x ; \quad$ Find $( f \circ g ) ( 0 )$ .

Find the domain of the function. - $f ( x ) = \ln \left( \frac { 1 } { x - 9 } \right)$

Solve the equation. - $243 ^ { 3 x - 1 } = 27 ^ { 2 x }$

Find the present value. Round to the nearest cent. -To get $10,000 after 3 years at 6% compounded monthly

The function f is one-to-one. Find its inverse. - $f ( x ) = 2 x + 8$

Find the effective rate of interest. - $6 \frac { 1 } { 4 } \%$ % compounded monthly

Solve the equation. - $4 ( 1 + 2 x ) = 1,024$

Solve the problem. -The logistic growth functi $f ( t ) = \frac { 440 } { 1 + 6.3 e ^ { - 0.2 t } }$ describes the population of a species of butterflies t months after they are introduced to a non-threatening habitat. What is the limiting size of the butterfly population that the Habitat will sustain?

Write as the sum and/or difference of logarithms. Express powers as factors. - $\ln \sqrt [ 6 ] { \mathrm { ey } }$

Solve the equation. - $3 ^ { 2 x } + 3 ^ { x } - 6 = 0$

Determine whether the given function is exponential or not. If it is exponential, identify the value of the base a. - () -1 3 0 7 1 11 2 15 3 19

Solve the problem. -Strontium 90 decays at a constant rate of 2.44% per year. Therefore, the equation for the amount P of strontium 90 after t years is P = P0 e-0.0244t. How long will it take for 15 grams of strontium to decay to 5 grams? Round Answer to 2 decimal places.

Change the exponential expression to an equivalent expression involving a logarithm. - $7 ^ { 3 } = x$

Solve the problem. -The formul $\mathrm { A } = 180 \mathrm { e } ^ { 0.036 \mathrm { t } }$ models the population of a particular city, in thousands, t years after 1998. When will the population of the city reach 287 thousand?

Find functions f and g so that f $f \circ g = H$ - $H ( x ) = \left| 4 - 3 x ^ { 2 } \right|$

The graph of a one-to-one function f is given. Draw the graph of the inverse function f-1 as a dashed line or curve. - $f ( x ) = \sqrt { x + 2 }$ A) B)

Solve the problem. -Conservationists tagged 130 black-nosed rabbits in a national forest in 1990. In 1,991, they tagged 260 black-nosed rabbits in the same range. If the rabbit population follows the exponential law, how many rabbits Will be in the range 9 years from 1990?

Solve the problem. -In 1992, the population of a country was estimated at 5 million. For any subsequent year, the population, P(t) (in millions), can be modeled using the equation $\mathrm { P } ( \mathrm { t } ) = \frac { 250 } { 5 + 44.99 \mathrm { e } ^ { - 0.0208 \mathrm { t } } }$ , where t is the number of years since 1992. Determine the year when the population will be 39 million.

Solve the problem. -Find a so that the graph of $f ( x ) = \log _ { a } x$ contains the point $( 4,10 )$ .

Find the value of the expression. -Let $\log _ { \mathrm { b } } \mathrm { A } = 2$ and $\log _ { \mathrm { b } } \mathrm { B } = - 10$ . Find $\log _ { \mathrm { b } } \frac { \mathrm { A } } { \mathrm { B } }$ .