Logarithms & Exponential Functions Cheat Sheet

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.

Understanding Logarithms & Exponential Functions

Exponential models are different from linear models because they change by a constant factor, not a constant amount. A savings balance with compound interest is multiplied by the same growth factor each period. A population can behave this way when each individual produces an average number of offspring.

The starting value sets the initial size of the model. The base controls how quickly the quantity changes. A base of one point zero five means a five percent increase for each time period.

A base of zero point eight means the quantity keeps eighty percent of its previous value. Decay approaches zero but does not normally reach zero at a finite time.

Logarithms are useful when the values cover a huge range. The Richter scale for earthquakes, decibel levels for sound, and pH measurements use logarithmic ideas. Equal steps on these scales represent multiplication by the same factor, rather than addition of the same amount.

This is why a small change in a reported logarithmic value can mean a large physical change. On a calculator, common logarithms use base ten and natural logarithms use base e. Either can be used to find a logarithm in another base through a ratio.

The important restriction is that the whole input to a logarithm must be positive. A negative number and zero do not have real logarithms.

When solving an exponential equation, first isolate the exponential part if possible. If both sides can be written using the same base, compare the exponents. Otherwise, take a logarithm of both sides and solve the resulting equation.

Keep track of units in word problems. If time is measured in months, an answer of three point five means three and a half months, not three months. For equations containing logarithms, combine or simplify the logarithms only when a valid law applies.

A common error is treating the logarithm of a sum as the sum of two logarithms. That rule does not exist. Always substitute final answers back into the original equation, since algebra steps can produce values that make a logarithm undefined.

Graph changes help make formulas easier to read. Multiplying an exponential function by a positive number stretches it vertically. Adding a number shifts the graph upward or downward and changes the horizontal level that the graph approaches.

For a logarithmic graph, shifts can move the vertical boundary where the graph is not defined. Inverse graphs exchange inputs and outputs, so an intercept or point on one graph becomes a reversed point on the other.

Make a small table of values when learning these graphs. Notice which values are allowed, where the curve changes slowly or quickly, and whether the context requires whole numbers or permits decimals.

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Logarithms & Exponential Functions Cheat Sheet

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Understanding Logarithms & Exponential Functions