Exponential & Logarithmic Equations & Inequalities (College Board AP® Precalculus): Exam Questions

The function $m$ is given by $m\left(x\right)=e^{2x}-e^x-12$. Find all input values in the domain of $m$ that yield an output value of 0.

The function $g$ is given by $g\left(x\right)=e^{\left(x+3\right)}$.

Solve $g\left(x\right)=10$ for values of $x$ in the domain of $g$.

The function $g$ is given by $g\left(x\right)=2.916·{\left(0.7\right)}^x$.

Find all values of $x$, as decimal approximations, for which $g\left(x\right)=2$, or indicate that there are no such values.

The function $f$ is given by $f\left(x\right)={\log }_{5}x$.

Solve $f\left(x\right)=3$ for values of $x$ in the domain of $f$.

The function $f$ is given by $f\left(x\right)=e^{x+1}$.

Find an expression for $f^{-1}\left(x\right)$. Show the work that leads to your answer.

The function $g$ is given by $g\left(x\right)={\log }_2\left(x+5\right)$.

Find an expression for $g^{-1}\left(x\right)$. Show the work that leads to your answer.

The function $g$ is given by $g\left(x\right)=\ln \left(x^2-9\right)$.

Solve $g\left(x\right)=\ln 40$ for values of $x$ in the domain of $g$.

The function $m$ is given by $m\left(x\right)=\frac{e^{4x}}{e^{4x}-12}$. Find all input values in the domain of $m$ that yield an output value of $4$.

The function $k$ is given by $k\left(x\right)=6e^{\left(2x\right)}-e$

Solve $k\left(x\right)=2e$ for values of $x$ in the domain of $k$.

The number of visitors to a museum, in thousands, $t$ years after it opened is modeled by the function $V\left(t\right)=a{\left(1.08\right)}^t$, where $a$ is a constant. Two years after the museum opened, there were $46.656$ thousand visitors.

Find the value of $a$. Then find the value of $t$, as a decimal approximation, for which $V\left(t\right)=100$.

The concentration of a medication in a patient's bloodstream, in milligrams per liter, $t$ hours after administration is modeled by the function $C\left(t\right)=12{\left(0.75\right)}^t$.

The medication is effective only when its concentration is at least $2$ milligrams per liter. Determine all values of $t$, as decimal approximations, for which the medication is effective. Show the work that leads to your answer.

The function $g$ is given by $g\left(x\right)={\log }_3\left(x-4\right)+2$.

Find an expression for $g^{-1}\left(x\right)$, and state the domain of $g^{-1}$. Show the work that leads to your answer.

The function $f$ is given by $f\left(x\right)=5-2·3^{x+1}$.

Find an expression for $f^{-1}\left(x\right)$, and state the domain of $f^{-1}$. Show the work that leads to your answer.