Logarithm Math Example 5

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

Example 5

hard
Solve log⁡2(x)+log⁡2(x−6)=4\log_2(x) + \log_2(x - 6) = 4.

Solution

  1. 1
    Apply the product rule for logarithms: log⁡2[x(x−6)]=4\log_2[x(x - 6)] = 4.
  2. 2
    Rewrite in exponential form: x(x−6)=24=16x(x - 6) = 2^4 = 16. Expand: x2−6x−16=0x^2 - 6x - 16 = 0.
  3. 3
    Factor: (x−8)(x+2)=0(x - 8)(x + 2) = 0, giving x=8x = 8 or x=−2x = -2.
  4. 4
    Check domain: log⁡2(x)\log_2(x) requires x>0x > 0 and log⁡2(x−6)\log_2(x - 6) requires x>6x > 6. So x=−2x = -2 is extraneous. The solution is x=8x = 8.

Answer

x=8x = 8
Logarithmic equations often reduce to polynomial equations via the product rule and exponential conversion. Always check that solutions satisfy the domain restrictions — logarithms are only defined for positive arguments.

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 4 medium

Solve [formula] where [formula] denotes [formula].

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