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

What are all values of $x$ for which $9^{x+2}={27}^x$?

What is the value of $x$ for which ${\log }_{2}x=5$?

The function $f$ is given by $f\left(x\right)=e^x-4$.

Which of the following is an expression for $f^{-1}\left(x\right)$?

The function $P$ is given by $P\left(t\right)=\frac{800,000}{1+0.6e^{kt}}$, where $k$ is a constant. If $P\left(5\right)=450,000$, what is the value of $P\left(15\right)$ ?

The function $f$ is given by $f\left(x\right)=4^{x-3}+5$. Which of the following is an expression for $f^{-1}\left(x\right)$?

The function $P$ is given by $P\left(t\right)=\frac{500,000}{1+0.5e^{kt}}$, where $k$ is a constant. If $P\left(6\right)=320,000$, what is the value of $P\left(18\right)$ ?

What are all values of $x$ for which $\ln \left(x^5\right)-\ln x=8$?

What are all values of $x$ for which ${\log }_{2}x+{\log }_2\left(x-3\right)=2$?

The function $h$ is given by $h\left(x\right)=5·{\left(1.4\right)}^x$. What are all values of $x$ for which $h\left(x\right)\ge 15$?

What are all values of $x$ for which $2\ln \left(x+1\right)<4$?

The temperature of a liquid, in degrees Celsius, $t$ minutes after being placed in a freezer is modeled by the function $T\left(t\right)=15·{\left(0.8\right)}^t-5$. What are all values of $t$ for which $T\left(t\right)=-10$, or indicate that there are no such values?

Consider the functions $f$ and $g$ given by $f\left(x\right)={\log }_3\left(x-2\right)+{\log }_3\left(x+4\right)$ and $g\left(x\right)={\log }_3\left(x+10\right)$. In the $xy$-plane, what are all $x$-coordinates of the points of intersection of the graphs of $f$ and $g$?

The function $f$ is given by $f\left(x\right)=-3e^{x-2}+4$. Let $f^{-1}$ be the inverse function of $f$. What is the domain of $f^{-1}$?

What are all values of $x$ for which $e^{2x}-9e^x+14=0$?

The function $g$ is given by $g\left(x\right)=4-2{\log }_3\left(x+5\right)$.

Which of the following is an expression for $g^{-1}\left(x\right)$?

Find all real solutions to $e^{2x}-5e^x+6=0$.

How many real values of $x$ satisfy the equation ${\log }_2\left(x\right)+{\log }_2\left(x-3\right)=2$?