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ALGEBRA · EXPONENTS & LOGARITHMS

Exponents & Logarithms

Logarithms answer the question: what power do I need?

16 min read 4 practice questions
Exponents describe repeated multiplication. A logarithm is the inverse: log⁡b(x)\log_b(x) asks “to what power must I raise bb to get xx?”
log⁡b(x)=y  ⟺  by=x\log_b(x) = y \iff b^y = x
The exponent rules translate directly into log rules:
log⁡b(MN)=log⁡bM+log⁡bNlog⁡b ⁣(MN)=log⁡bM−log⁡bNlog⁡b(Mk)=klog⁡bM\log_b(MN) = \log_b M + \log_b N \qquad \log_b\!\left(\tfrac{M}{N}\right) = \log_b M - \log_b N \qquad \log_b(M^k) = k\log_b M

Worked example · Solving an exponential equation

Solve 3x=813^{x} = 81.
  1. 1
    Rewrite 8181 as a power of 33: 81=3481 = 3^4.
  2. 2
    So 3x=343^x = 3^4, and x=4x = 4.
  3. 3
    Equivalently, x=log⁡381=4x = \log_3 81 = 4.

CHECK YOUR UNDERSTANDING

Practice

Answer each question, then check.

1
Evaluate log⁡232\log_2 32.
2
Simplify log⁡39+log⁡327\log_3 9 + \log_3 27.
3
Solve 2x+1=642^{x+1} = 64.
4
Which is equivalent to log⁡(x3)\log(x^3)?

Numeric answers accept decimals or fractions like 3/4.