How to Solve Equations Involving Logarithms

In math, a logarithm is the inverse function of an exponential function. For example, if b^x = y, then x = Log(y) in base b. Logarithms are used to solve algebra equations that involve exponents, and conversely, exponents are used to solve equations that involve logarithms. If you encounter a logarithm equation in an algebra or precalculus class, here are the steps to solve the equation.

Difficulty: Moderately Easy

Instructions

1. First, apply the 4 basic rules of logs to simplify the equation. The 4 rules hold true regardless of the base of the logarithm.
   
   (1) log(a) + log(b) = log(ab)
   (2) log(a) - log(b) = log(a/b)
   (3) log(a^n) = n[log(a)]
   (4) log(1) = 0
   
   For example, if you equation is log(x) + log(x+3) = 1, it can be rewritten as log(x²+3x) = 1 by using the first rule.

2. Perform the same operations to both sides of the equation so that all of the terms with variables are on one side. For example, the equation
   2log(x) + log(5) = log(x) + 3
   can be simplified to
   2log(x) - log(x) = 3 - log(5)
   log(x) = 3 - log(5).

3. Use the appropriate base to cancel out the logarithms. Conventionally, the abbreviation "log" denotes logarithm base 10, and "ln" denotes logarithm base e, where e ≈ 2.71828. If the logarithm is with another base, that number will be written as a subscript below the "g" in the abbreviation "log."

4. To illustrate steps 1 through 3, consider the following example:
   2log(x) - log(x-2.5) = 1.
   
   First, apply rule three to make the left side into
   log(x²) - log(x-2.5) = 1.
   
   Then, apply rule two to obtain
   log(x²/(x-2.5)) = 1.
   
   Now make both sides into exponents base 10:
   10^[log(x²/(x-2.5))] = 10^1
   x²/(x-2.5) = 10.
   
   And now use regular algebra to solve.
   x²/(x-2.5) = 10
   x² = 10x - 25
   x² - 10x + 25 = 0
   (x-5)² = 0
   x = 5.