Logarithmic Functions (College Board AP® Precalculus): Exam Questions

What is the exact value of ${\log }_{2}32$?

What is the exact value of $\ln \left(e^4\right)$?

For $x>0$, which of the following is equivalent to $e^{\ln \left(5x\right)}$?

The graph of the exponential function $g\left(x\right)=b^x$, where $b>0$ and $b\ne 1$, passes through the point $\left(3,8\right)$. Which of the following points must lie on the graph of the logarithmic function $f\left(x\right)={\log }_{b}x$?

Let $m$, $n$, and $p$ be positive constants. Which of the following is equivalent to ${\log }_{10}\left(\frac{m^3}{n{p}^2}\right)$?

Let $a$ and $b$ be positive constants. Which of the following is equivalent to $3\ln a+2\ln b$?

The function $f$ is given by $f\left(x\right)={\log }_3\left(x-4\right)+2$. Which of the following describes the domain and the vertical asymptote of $f$?

On a base-10 logarithmic scale, the measurement of quantity $X$ is $3.2$ and the measurement of quantity $Y$ is $5.6$. Which of the following is the best approximation for how many times greater the actual value of quantity $Y$ is than the actual value of quantity $X$?

Let $x$ and $y$ be positive constants. Which of the following is equivalent to ${\log }_2\left(\frac{x^4}{y^3}\right)$?

The exponential function $f$ is given by $f\left(x\right)=5^x$. The function $f$ can be rewritten in the equivalent form $f\left(x\right)=e^{kx}$, where $k$ is a constant. What is the value of $k$, expressed as a decimal approximation?

The function $f$ is given by $f\left(x\right)={\log }_3\left(x-h\right)+k$, where $h$ and $k$ are constants. The graph of $f$ has an $x$-intercept at $x=4$ and a vertical asymptote at $x=1$.

What is the value of $f\left(28\right)$?

The decibel scale measures sound intensity by the formula $L=10{\log }_{10}\left(\frac{I}{I_0}\right)$, where $I$ is the sound intensity, $I_0$ is a reference intensity, and $L$ is the sound level in decibels.

Sound A has a sound level of $60$ decibels. Sound B has an intensity that is $1000$ times greater than the intensity of sound A.

What is the sound level of sound B in decibels?

The pH scale measures the acidity of a solution by the formula $\text{pH}=-{\log }_{10}\left[H^{+}\right]$, where $\left[H^{+}\right]$ is the hydrogen ion concentration in moles per liter.

Solution X has a pH of $4.5$. Solution Y has a hydrogen ion concentration that is $100$ times the hydrogen ion concentration of solution X.

What is the pH of solution Y?

If $x>1$, which of the following is equivalent to $2{\log }_2\left(x\right)-\frac{1}{2}{\log }_2\left(x^2-1\right)+{\log }_2\left(3\right)$?

Consider the function $f$ defined by $f\left(x\right)=-{\log }_3\left(x-2\right)+1$. Which of the following statements about $f$ is true?