In the previous section, we mentioned that calculators and computers can typically evaluate logs for two bases directly, 10 and e. This section shows how to evaluate logs for a base other than 10 or e (as long as the base is a non-negative number other than 1).

Solve by observation

• For the equation x = 2^y, find y if x = 16.
  • First, take the base-2 log of both sides: log₂(x) = log₂(2^y) = y.
  • Next, we want to evaluate y = log₂(16).
  • If we consider the powers of 2 (i.e., 2, 4, 8, 16, 32, …), we know that 2⁴ = 16, so y = log₂(16) = 4.

Solve by changing the base

• For the equation x = 2^y, find y if x = 20.
  • First, let’s try taking the base-2 log of both sides: log₂(x) = log₂(2^y) = y.
  • Next, we want to evaluate y = log₂(20).
  • There is no integer power of 2 that produces 20, so solving by observation won’t work.
  • Instead, go back to the original equation, x = 2^y, and take the common log of both sides: log₁₀(x) = log₁₀(2^y) = ylog₁₀(2) (by the exponent property of logs).
  • Solve for y: y = log₁₀(x)/log₁₀(2).
  • Finally, evaluate y at x = 20 using a calculator or computer: y = log₁₀(20)/log₁₀(2) = 4.3219.
  • Check: 2^4.3219 = 20.00.

• In the previous example, we used the common log, but either the common log or natural log work. In this example, let’s use the natural log. For the equation x = 5^y, find y if x = 20.
  • First, take the natural log of both sides: ln(x) = ln(5^y) = yln(5) (by the exponent property of logs).
  • Solve for y: y = ln(x)/ln(5).
  • Finally, evaluate y at x = 20 using a calculator or computer: y = ln(20)/ln(5) = 1.8614.
  • Check: 5^1.8614 = 20.00.