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

The function $j$ is given by $j\left(x\right)={\log }_{10}\left(8x^5\right)+{\log }_{10}\left(2x^2\right)-9{\log }_{10}x$.

Rewrite $j\left(x\right)$ as a single logarithm base 10 without negative exponents in any part of the expression. Your result should be of the form ${\log }_{10}\left(\mathrm{expression}\right)$.

The function $f$ is given by $f\left(x\right)=\ln x$.

Solve $f\left(x\right)=9$ for values of $x$ in the domain of $f$.

The function $g$ is given by $g\left(x\right)={\log }_3\left(3^x\right)$.

Rewrite $g\left(x\right)$ as an expression that does not involve a logarithm.

The function $h$ is given by $h\left(x\right)=e^{x\ln 4}$.

Rewrite $h\left(x\right)$ as an equivalent expression of the form $h\left(x\right)=b^x$, where $b$ is a constant. Identify the value of $b$.

The function $j$ is given by$j\left(x\right)=3{\log }_5\left(2x\right)-\frac{1}{3}{\log }_{5}x$

Rewrite $j\left(x\right)$ as a single logarithm base 5 without negative exponents in any part of the expression. Your result should be of the form ${\log }_5\left(\mathrm{expression}\right)$.

The function $g$ is given by $g\left(x\right)=2\ln x+\frac{1}{3}\ln x$.

Rewrite $g\left(x\right)$ as a single logarithm without negative exponents in any part of the expression. Your result should be in the form $\ln \left(\mathrm{expression}\right)$.

The function $f$ is given by $f\left(x\right)=6^x$.

Rewrite $f\left(x\right)$ in the equivalent form $f\left(x\right)=e^{kx}$. Express the exact value of $k$ using a natural logarithm, then express $k$ as a decimal approximation.

The function $g$ is given by $g\left(x\right)=-3{\log }_2\left(x-4\right)+1$.

State whether the function $g$ is increasing or decreasing on its domain. Then determine the end behavior of $g$ as $x$ increases without bound. Express your answer using the mathematical notation of a limit.

On a base-10 logarithmic scale measuring earthquake magnitudes, Earthquake A has a measurement of $4.5$ and Earthquake B has a measurement of $6.2$.

Determine how many times greater the actual amplitude of Earthquake B is than the actual amplitude of Earthquake A. Express your answer as a decimal approximation. Show the work that leads to your answer.

The functions $f$ and $g$ are given by $f\left(x\right)=5^x$ and $g\left(x\right)={\log }_{5}x$.

Determine the exact value of $f\left(g\left(14\right)\right)$.

Determine the exact value of ${\log }_4\left(\frac{1}{8}\right)$. Show the work that leads to your answer.

The function $h$ is given by $h\left(x\right)={\log }_{7}x$.

Rewrite $h\left(20\right)$ using the change of base property with the natural logarithm. Then evaluate the expression as a decimal approximation.

The function $j$ is given by $j\left(x\right)={\log }_2\left(x^4\right)-3{\log }_2\left(\sqrt{x}\right)+2{\log }_2\left(8x\right)$, defined for $x>0$.

Rewrite $j\left(x\right)$ as a single logarithm base 2 without negative exponents in any part of the expression. Your result should be of the form ${\log }_2\left(c{x}^n\right)$, where $c$ and $n$ are constants.

The function $f$ is given by $f\left(x\right)=\ln \left(e^{2x+1}\right)+\ln \left(e^{x-3}\right)-2\ln \left(\sqrt{e^x}\right)$, defined for all real numbers.

Rewrite $f\left(x\right)$ as a single linear expression in $x$.

The function $g$ is given by $g\left(x\right)=2{\log }_5\left(3x+1\right)-4$, defined for $x>-\frac{1}{3}$.

Let $g^{-1}$ be the inverse function of $g$. Find the value of $g^{-1}\left(2\right)$ as a decimal approximation, or indicate that it is not defined.

Find all values of $x$ for which $g\left(x\right)=0$, as a decimal approximation, or indicate there are no such values.