Logarithms & Exponential Functions Cheat Sheet

A printable reference covering exponential rules, logarithm definitions, log laws, graph features, solving equations, and change of base for grades 10-11.

Download PNG

Logarithms and exponential functions describe situations where quantities grow, shrink, or are measured by powers. This cheat sheet helps students connect exponential notation, logarithmic notation, graphs, and equation-solving methods. It is useful for simplifying expressions, solving growth and decay problems, and checking domain restrictions. The main idea is that logarithms undo exponentials, so each form gives a different view of the same relationship. The core connection is $a^x=y$ if and only if $\log_a y=x$ , where $a>0$ and $a\ne1$ . Exponential functions often have the form $f(x)=ab^x$ , while logarithmic functions often have the form $f(x)=\log_b x$ . Log laws turn multiplication into addition, division into subtraction, and powers into products. Graph features such as asymptotes, intercepts, domain, range, and inverse relationships help students interpret answers, not just calculate them.

Key Facts

For $a>0$ and $a\ne1$ , $a^x=y$ if and only if $\log_a y=x$ , so a logarithm gives the exponent needed to make $y$ .
The product rule is $\log_b(MN)=\log_b M+\log_b N$ , where $M>0$ , $N>0$ , $b>0$ , and $b\ne1$ .
The quotient rule is $\log_b\left(\frac{M}{N}\right)=\log_b M-\log_b N$ , where $M>0$ and $N>0$ .
The power rule is $\log_b(M^p)=p\log_b M$ , which moves an exponent on the argument to the front as a multiplier.
The change of base formula is $\log_b M=\frac{\log_a M}{\log_a b}$ , so a calculator can evaluate logs in any valid base.
An exponential function $f(x)=ab^x$ grows when $b>1$ and decays when $0<b<1$ .
The functions $y=b^x$ and $y=\log_b x$ are inverses, so their graphs reflect across the line $y=x$ .
The graph of $y=\log_b x$ has domain $x>0$ , range all real numbers, and vertical asymptote $x=0$ .

Vocabulary

Exponential function: An exponential function has the variable in the exponent, such as $f(x)=ab^x$ , where $a\ne0$ , $b>0$ , and $b\ne1$ .
Logarithm: A logarithm is the exponent needed to produce a number, so $\log_b x=y$ means $b^y=x$ .
Base: The base is the repeated factor in an exponential expression, such as $b$ in $b^x$ or in $\log_b x$ .
Argument: The argument of a logarithm is the input being logged, such as $x$ in $\log_b x$ , and it must be positive.
Asymptote: An asymptote is a line that a graph approaches but does not touch, such as $x=0$ for the parent graph $y=\log_b x$ .
Inverse functions: Inverse functions undo each other, so $y=b^x$ and $y=\log_b x$ reverse inputs and outputs.

Common Mistakes to Avoid

Forgetting domain restrictions for logarithms is wrong because $\log_b x$ is defined only when $x>0$ , so possible solutions that make an argument nonpositive must be rejected.
Writing $\log_b(M+N)=\log_b M+\log_b N$ is wrong because the product rule applies to multiplication, not addition.
Dropping the base of a logarithm is wrong because $\log_2 8$ and $\log_{10} 8$ have different values.
Solving $b^x=b^y$ without checking the bases is wrong because $x=y$ follows directly only when both sides have the same valid base $b>0$ and $b\ne1$ .
Confusing exponential and logarithmic asymptotes is wrong because $y=b^x$ has horizontal asymptote $y=0$ , while $y=\log_b x$ has vertical asymptote $x=0$ .

Practice Questions

1 Rewrite $3^4=81$ in logarithmic form.
2 Evaluate $\log_2 32$ and explain what exponent it represents.
3 Solve $5^{x}=125$ and then solve $\log_3(x-1)=2$ .
4 Explain why $y=2^x$ and $y=\log_2 x$ are inverse functions, and describe how their graphs are related.