(rev. 6/16/2000)

General Result:
If b is any legitimate base of a logarithm (such as 10, or e), then log_a x = (log_b x) / (log_b a).

Examples of How This Can Be Useful:
Example 1. Compute log_p 17.
Answer: On your calculator, punch in log(17)/log(3.14159) ENTER. The answer is approximately 2.475.
Alternate Answer: Punch in ln(17)/ln(3.14159) ENTER. Again, the answer is approximately 2.475. The Change-of-Base Formula allows you to use any new base as long as you use it consistently. In practice, you will always use either 10 or e for your new base, since those are the only bases for which your calculator provides a log key.

Example 2. Compute log₁₃₈ [(7.5 ´ 10^–38) / (3.28 ´ 10⁹³)].
Answer: First, rewrite this as log₁₃₈ (7.5 ´ 10^–38) – log₁₃₈ (3.28 ´ 10⁹³). By the Change-of-Base Formula, we have ln(7.5 ´ 10^–38)/ln(138) – ln(3.28 ´ 10⁹³)/ln(138) = –16.88176634 – 44.1687918 = –61.051, correct to 3 decimal places.
Alternate Answer: Practice getting the same answer by using common logs instead of natural logs.
Yet Another Method: Use algebra to rewrite the problem as log₁₃₈ [7.5/3.28 ´ 10^{–38 – 93}] = log₁₃₈ [2.286585366 ´ 10^–131] = log₁₃₈ 2.286585366 + log₁₃₈ 10^–131 = (log 2.286585366)/(log 138) + (log 10^–131)/(log 138) = 0.3591874197/2.139879086 + (–131/2.139879086) = (0.3591874197 – 131)/2.139879086 = –61.051, correct to 3 decimal places.

Proof of the General Result:
Given: b > 0, b ¹ 1
Prove: log_a x = (log_b x) / (log_b a)
Proof: Since we don’t yet know what the value of log_a x is, let y denote the answer. We have y = log_a x. By definition of logarithm, y is the exponent we place on a in order to obtain x. That is, a^y = x. Since b > 0 and b ¹ 1, b can be used as a base of a logarithm, and we can take the base b logarithm of both sides to get log_b a^y = log_b x. By the exponent rule for logarithms, the LHS can be written as y log_b a, giving us the equation y log_b a = log_b x. Divide through by log_b a to get y = (log_b x) / (log_b a).

Q.E.D.