Logarithms help students solve equations where the variable is in an exponent. This cheat sheet summarizes the main logarithm properties used in Algebra 2, Precalculus, and advanced high school math. Students need these rules to simplify expressions, solve exponential equations, and understand inverse functions.

It is a quick reference for choosing the correct property without mixing up operations.

The most important idea is that log⁡b(x)\log_b(x) answers the question, bb raised to what power equals xx. Logarithms turn multiplication into addition, division into subtraction, and powers into multiplication. The change of base formula lets students evaluate logs with bases not available on a calculator.

Domain restrictions matter because log⁡b(x)\log_b(x) is defined only when b>0b>0, b≠1b\ne 1, and x>0x>0.

Key Facts

  • The logarithmic equation log⁡b(x)=y\log_b(x)=y is equivalent to the exponential equation by=xb^y=x, where b>0b>0, b≠1b\ne 1, and x>0x>0.
  • The product property is log⁡b(MN)=log⁡b(M)+log⁡b(N)\log_b(MN)=\log_b(M)+\log_b(N) for M>0M>0 and N>0N>0.
  • The quotient property is log⁡b(MN)=log⁡b(M)−log⁡b(N)\log_b\left(\frac{M}{N}\right)=\log_b(M)-\log_b(N) for M>0M>0 and N>0N>0.
  • The power property is log⁡b(Mp)=plog⁡b(M)\log_b(M^p)=p\log_b(M) for M>0M>0.
  • The change of base formula is log⁡b(M)=log⁡a(M)log⁡a(b)\log_b(M)=\frac{\log_a(M)}{\log_a(b)}, where a>0a>0, a≠1a\ne 1, b>0b>0, and b≠1b\ne 1.
  • The inverse properties are log⁡b(bx)=x\log_b(b^x)=x and blog⁡b(x)=xb^{\log_b(x)}=x for x>0x>0.
  • The special values are log⁡b(1)=0\log_b(1)=0 and log⁡b(b)=1\log_b(b)=1 for b>0b>0 and b≠1b\ne 1.
  • A logarithm with base 1010 is written log⁡(x)\log(x), and a logarithm with base ee is written ln⁡(x)\ln(x).

Vocabulary

Logarithm
A logarithm log⁡b(x)\log_b(x) is the exponent that base bb must be raised to in order to get xx.
Base
The base bb in log⁡b(x)\log_b(x) is the number being raised to a power, and it must satisfy b>0b>0 and b≠1b\ne 1.
Argument
The argument is the input xx in log⁡b(x)\log_b(x), and it must be positive.
Common Logarithm
A common logarithm is a logarithm with base 1010, written as log⁡(x)\log(x).
Natural Logarithm
A natural logarithm is a logarithm with base ee, written as ln⁡(x)\ln(x).
Change of Base
Change of base rewrites log⁡b(M)\log_b(M) as log⁡a(M)log⁡a(b)\frac{\log_a(M)}{\log_a(b)} so it can be evaluated using another base.

Common Mistakes to Avoid

  • Writing log⁡b(M+N)=log⁡b(M)+log⁡b(N)\log_b(M+N)=\log_b(M)+\log_b(N) is wrong because the product property works only for multiplication, not addition.
  • Writing log⁡b(MN)=log⁡b(M)log⁡b(N)\log_b\left(\frac{M}{N}\right)=\frac{\log_b(M)}{\log_b(N)} is wrong because division inside a logarithm becomes subtraction, so the correct rule is log⁡b(M)−log⁡b(N)\log_b(M)-\log_b(N).
  • Forgetting domain restrictions is wrong because log⁡b(x)\log_b(x) is defined only when b>0b>0, b≠1b\ne 1, and x>0x>0.
  • Using the power property as log⁡b(Mp)=(log⁡b(M))p\log_b(M^p)=\left(\log_b(M)\right)^p is wrong because the exponent becomes a coefficient, so log⁡b(Mp)=plog⁡b(M)\log_b(M^p)=p\log_b(M).
  • Canceling incorrectly in expressions like log⁡b(bx)\log_b(bx) is wrong because log⁡b(bx)=log⁡b(b)+log⁡b(x)=1+log⁡b(x)\log_b(bx)=\log_b(b)+\log_b(x)=1+\log_b(x), not xx.

Practice Questions

  1. 1 Rewrite log⁡3(81)=4\log_3(81)=4 as an exponential equation.
  2. 2 Expand log⁡2(8x3y)\log_2\left(\frac{8x^3}{y}\right) using logarithm properties, assuming x>0x>0 and y>0y>0.
  3. 3 Solve log⁡5(x)=3\log_5(x)=3 for xx.
  4. 4 Explain why log⁡b(M+N)\log_b(M+N) cannot be expanded as log⁡b(M)+log⁡b(N)\log_b(M)+\log_b(N), and describe which operation the product property actually applies to.

Understanding Logarithm Properties Reference

A useful way to understand logarithms is through the shape of their graphs. For a base greater than one, the logarithm graph rises slowly as the input increases. It passes through the point where the input is one and the output is zero.

It never touches the vertical axis because zero and negative inputs are not allowed. This graph is the reflection of the matching exponential graph across the line where output equals input.

That reflection shows why each operation reverses the other. It also helps explain why a large change in an input may create only a small change in its logarithm.

The rules come from exponent laws, not from a pattern to memorize. Suppose two positive numbers are written as powers of the same base. Multiplying those numbers adds their exponents, so the logarithm of a product becomes a sum.

Dividing subtracts exponents, so the logarithm of a quotient becomes a difference. Raising a number to a power multiplies its exponent, which creates the power rule. This origin gives students a way to rebuild a forgotten rule.

It also exposes a major mistake. A logarithm of a sum does not split into two logarithms. Addition inside a logarithm has no simple expansion rule.

When solving a logarithmic equation, first isolate the logarithm if possible. Then rewrite the statement in exponential form and solve the resulting equation. Every proposed answer must be checked in the original expression.

Algebra can produce a value that makes a logarithm input zero or negative, and that value must be rejected. Expressions need attention before applying a property too.

For example, a difference of logarithms can combine into one logarithm of a quotient, but the numerator and denominator must stay positive. Parentheses matter because a coefficient in front of a logarithm is not the same as an exponent outside the logarithm.

Students meet logarithmic scales whenever values span an enormous range. The pH scale compares acidity through powers of ten. Sound intensity in decibels uses a logarithmic comparison because human hearing responds to ratios of intensities.

Earthquake magnitude, stellar brightness, computer data measures, and compound interest models can involve logarithms. In these settings, an increase of one unit often means multiplication by a fixed factor rather than adding one ordinary unit. Calculator work needs care as well.

The common log key uses base ten, while the natural log key uses the base called e. Either key can evaluate another base by forming a ratio of two logs with the same calculator key.