0. Functions

Which operation can be used to eliminate the natural logarithm ($\ln$) from both sides of an equation such as $\ln \left(x\right)=3$?

Given the following table of values for an exponential function $f\left(x\right)$:

Which of the following transformations results in a vertical stretch of the exponential decay function $f\left(x\right)=e^{-x}$?

A culture of bacteria has a population of $150$ cells when it is first observed. The population doubles every $12~\text{hr}$, which means its population is governed by the function $p\left(t\right)=150\cdot{2^{\frac{t}{12}}}$, where $t$ is the number of hours after the first observation.

What is the population $4~\text{days}$ after the first observation?

Which of the following best describes the function $Ei\left(x\right)$?

Determine if the function is an exponential function.  

If so, identify the power & base, then evaluate for $x=4$.

$f\left(x\right)=\left(-2\right)^{x}$

Determine if the function is an exponential function.  

If so, identify the power & base, then evaluate for $x=4$ .

$f\left(x\right)=3\left(1.5\right)^{x}$

Determine if the function is an exponential function.  

If so, identify the power & base, then evaluate for $x=4$ .

$f\left(x\right)=\left(\frac12\right)^{x}$

Graph the given function.

$g\left(x\right)=e^{x+3}-1$

A culture of bacteria has a population of $150$ cells when it is first observed. The population doubles every $12~\text{hr}$, which means its population is governed by the function $p\left(t\right)=150\cdot{2^{\frac{t}{12}}}$, where $t$ is the number of hours after the first observation.

How long does it take the population to triple in size?

A culture of bacteria has a population of $150$ cells when it is first observed. The population doubles every $12~\text{hr}$, which means its population is governed by the function $p\left(t\right)=150\cdot{2^{\frac{t}{12}}}$, where $t$ is the number of hours after the first observation.

How long does it take the population to reach $10,000$?

Solve each equation.

$48=6e^{4k}$

Solve each equation.

$7^{y-3}=50$

Write the following logarithms in terms of the natural logarithm. Then use a calculator to find the value of the logarithm, rounding your result to four decimal places.

${\log }_{6}60$

Changing bases Convert the following expressions to the indicated base.

$\ln \mid x\mid$ using base 5