Exam 5: Exponential and Logarithmic Functions

Write $5 \ln ( x + 7 ) - 3 \ln x - 4 \ln \left( x ^ { 2 } + 8 \right)$ as a single logarithm with a coefficient of 1. Assume all variable expressions represent positive real numbers.

Find the domain of the function $k ( x ) = \log _ { 8 } ( 4 - x )$ .

Sketch the graph of the function $N ( t ) = 2 - e ^ { t }$

Rewrite the logarithmic equation $\log _ { 6 } \frac { 1 } { 36 } = - 2$ in exponential form.

Solve for x: $5 ^ { - x / 3 } = 0.0052$ . Round to 3 decimal places.

Find the inverse function of the function f given by the set of ordered pairs. $\{ ( 6,2 ) , ( 5,3 ) , ( 4,4 ) , ( 3,5 ) \}$

Sketch the graph of the function $y = 3 ^ { - x ^ { 2 } }$ .

Expand the given logarithmic expression. Assume all variable expressions represent positive real numbers. $\ln \left( x ^ { 7 / 4 } y ^ { 3 / 2 } \right)$

Use the functions given by $f ( x ) = x + 4$ and $g ( x ) = 2 x - 6$ to find the composition of functions $(g\circ f)^{-1}$

Solve $\left( \frac { 1 } { 2 } \right) ^ { x } = 8$ for x.

Show that $f$ and $g$ are functions by using the definition of inverse functions. $f ( x ) = 5 x + 1 , g ( x ) = \frac { x - 1 } { 5 }$

Determine whether the function has an inverse function, If it does, find its inverse function. $f ( x ) = x ^ { 4 }$

Show that $f$ and $g$ are functions by using the definition of inverse functions. $f ( x ) = \frac { 1 } { x } , g ( x ) = \frac { 1 } { x }$

What is the half-life of a radioactive substance if 3.4 g decays to 0.80 g in 75 hours? Round to the nearest tenth of an hour.