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Change of Base Formula
Logarithmic functions are critical for describing exponential scaling like earthquake magnitudes or pH values. The change of base formula allows you to evaluate any logarithm on standard calculators. For instant computation, use our log calculator or check logarithmic progressions using the logarithm calculator.
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What is the change of base formula?
The change of base formula allows you to write any logarithm log_b(x) in terms of a different base d. It states that:
log_b(x) = log_d(x) / log_d(b)
Most commonly, the base is changed to 10 (common logarithm) or e (natural logarithm) so it can be evaluated directly on any standard computer or calculator:
Using Base 10 (log):
log_b(x) = log(x) / log(b)
Using Base e (ln):
log_b(x) = ln(x) / ln(b)
Variables Explained
- log_b(x) = The original logarithm you wish to evaluate.
- b = The original base of the logarithm.
- x = The argument of the logarithm (value inside).
- d = The new base (usually 10 or e).
Worked Examples
Example 1: Exact Log Evaluation (Base 2)
Evaluate log_2(8) using base 10 common logarithms.
- Formula: log_b(x) = log(x) / log(b)
- Given: b = 2, x = 8
- Calculation: log_2(8) = log(8) / log(2)
- Decimals: = 0.90309 / 0.30103
- Result: log_2(8) = 3 (since 2^3 = 8)
Example 2: Decimal Log Evaluation (Base 3)
Evaluate log_3(20) using natural logarithms (ln).
- Formula: log_b(x) = ln(x) / ln(b)
- Given: b = 3, x = 20
- Calculation: log_3(20) = ln(20) / ln(3)
- Decimals: = 2.99573 / 1.09861
- Result: log_3(20) = 2.7268
Common Mistakes with Change of Base
- Dividing before taking the logarithm: A common mistake is dividing the arguments first, e.g. calculating log(x / b). Note that log(x) / log(b) is not equal to log(x/b) (which is actually log(x) - log(b)).
- Swapping Numerator and Denominator: Be careful not to write the formula upside down, i.e., putting the base logarithm in the numerator log(b) / log(x). The log of the base always goes in the denominator (bottom).
- Mixing bases within a single equation: You must use the same base for both the top and bottom. For example, evaluating log(x) / ln(b) will yield an incorrect result.
Run the numbers
Solve general or custom base logarithms instantly using our calculators:
Frequently asked questions
The change of base formula is log_b(x) = log_d(x) / log_d(b). It allows you to rewrite any logarithm with a new base (d) so that it can be evaluated easily.
Most scientific and graphing calculators only have dedicated keys for base-10 logarithms (log) and base-e natural logarithms (ln). To evaluate a logarithm with a custom base like 2 or 5, you must rewrite it in terms of base-10 or base-e.
No. The ratio will remain exactly the same. For example, log_3(20) is equal to log(20)/log(3) and also equal to ln(20)/ln(3). Both yield the same decimal result.
No. Logarithm bases must always be positive real numbers other than 1. Therefore, the new base (d) must also be positive and not equal to 1.