Logarithm Properties Math Example 1

Follow the full solution, then compare it with the other examples linked below.

Example 1

easy
Expand log⁡2(8x3)\log_2(8x^3) using logarithm properties.

Solution

  1. 1
    Apply the product rule: log⁡2(8x3)=log⁡28+log⁡2x3\log_2(8x^3) = \log_2 8 + \log_2 x^3.
  2. 2
    Evaluate log⁡28=3\log_2 8 = 3 and apply the power rule: log⁡2x3=3log⁡2x\log_2 x^3 = 3\log_2 x.
  3. 3
    Result: 3+3log⁡2x3 + 3\log_2 x.

Answer

3+3log⁡2x3 + 3\log_2 x
The product rule log⁡b(MN)=log⁡bM+log⁡bN\log_b(MN) = \log_b M + \log_b N and power rule log⁡b(Mp)=plog⁡bM\log_b(M^p) = p\log_b M are the two most commonly used expansion properties.

About Logarithm Properties

The three fundamental rules of logarithms: the product rule log⁡b(xy)=log⁡bx+log⁡by\log_b(xy) = \log_b x + \log_b y, the quotient rule log⁡b ⁣(xy)=log⁡bx−log⁡by\log_b\!\left(\frac{x}{y}\right) = \log_b x - \log_b y, and the power rule log⁡b(xn)=nlog⁡bx\log_b(x^n) = n\log_b x.

Learn more about Logarithm Properties →

More Logarithm Properties Examples

Example 2 medium

Condense [formula] into a single logarithm.

Example 3 medium

Use the change of base formula to evaluate [formula] to three decimal places.

Example 4 easy

Expand [formula].

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