The base of a logarithm can be converted to another value through a
simple, one­step process.
LEARNING OBJECTIVE [ edit ]
Use the change of base formula to convert logarithms to base 10
KEY POINTS [ edit ]
If a logarithm and/or its base are not whole numbers, evaluation can be near­impossible.
One can change the base of a logarithm by expressing it as thequotient of two logarithms with a
common, same base.
Changing a logarithm's base to 10 makes it much simpler to evaluate; it can be done on a
calculator.
TERMS [ edit ]
base
A number raised to the power of an exponent.
logarithm
The logarithm of a number is the exponent by which another fixed value, the base, has to be
raised to produce that number.
Give us feedback on this content: FULL TEXT [ edit ]
So long as alogarithm, its base, and the number upon which it operates are allwhole
numbers, one can evaluate logarithms manually with minimal difficulty.
When decimals are involved, however, it can become exceedingly difficult to evaluate a
logarithm. Let's consider:
log4 (9)
We can easily determine that the above
will simplify to a number between one and
two, because 41 = 4 and 42 = 16. The exact
value, however, is not so easily
determined.
Not all calculators have
logarithm functions and, those that do
almost always have a base of 10.
Fortunately, any logarithm can be
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converted into a logarithm of equal value
with a different base. The formula for this transformation is:
loga (x) =
logb (x)
logb (a)
where a is the original base and b is the desired base.
Revisiting the example above, we can change the base from 4 to 10, which can be input into a
calculator.
log4 (9) =
log10 (9)
log10 (4)
The left side of the equation is extremely difficult to calculate manually, and would be
impossible to find on most calculators. However, the quotient of logarithms with base equal
to 10 can easily be found on a scientific calculator .
Interactive Graph: Graph of Binary Logarithm
Graph of a binary logarithm, y = 2ln(x) . This function can be expressed as the quotient of log (x) and
log (2), both with a base of 10.