Logarithm Math Example 4

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

Example 4

medium
Solve log⁡(x)+log⁡(x−3)=1\log(x) + \log(x - 3) = 1 where log⁡\log denotes log⁡10\log_{10}.

Solution

  1. 1
    Use the product rule: log⁡[x(x−3)]=1\log[x(x - 3)] = 1, so x(x−3)=101=10x(x - 3) = 10^1 = 10.
  2. 2
    Expand: x2−3x−10=0x^2 - 3x - 10 = 0. Factor: (x−5)(x+2)=0(x - 5)(x + 2) = 0, giving x=5x = 5 or x=−2x = -2.
  3. 3
    Since logarithms require positive arguments, x=−2x = -2 is extraneous. Thus x=5x = 5.

Answer

x=5x = 5
When solving logarithmic equations, always check that solutions keep all logarithm arguments positive. Extraneous solutions are common.

About Logarithm

The logarithm log⁡b(x)\log_b(x) answers: "to what power must bb be raised to produce xx?" It is the inverse function of f(x)=bxf(x) = b^x.

Learn more about Logarithm →

More Logarithm Examples

Example 1 easy

Evaluate [formula].

Example 2 medium

Solve [formula].

Example 3 easy

Evaluate [formula].

Example 5 hard

Solve [formula].

← All Logarithm Examples Logarithm Concept