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Change of Base Rule

The change of base rule for logarithms is:

log a (x) = log b (x) / log b (a)   a,b,x ∈ R + a,b ≠ 1

R + denotes the set of positive real numbers.

It is important to understand that b can be any valid base including, but not limited, to 2,10 and e.

Change of Base Rule Proof

b x = a ⇒ x = log b(a)

We take the logarithm base d on both sides:

log d(b) x = log d(a)

We apply the logarithm power rule:

log d(b) x = x log d(b)

x log d(b) = log d(a) ⇒ x = log d(a) / log d(b)

Because of our assumption:

x = log b(a) ⇒ log b(a) = log d(a) / log d(b)


  1. b = 2
    Log a (x) = log 2 (x) / log 2 (a)
    Log 4 (16) = log 2 (16) / log 2 (4)
    = 4 / 2 = 2
  2. b = 10
    Log a (x) = log (x) / log (a)
    Log 4 (16) = log (16) / log (4)
    = 1.2041199826 / 0.6020599913 = 2
  3. b = e
    Log a (x) = ln (x) / ln (a)
    Log 4 (16) = ln(16) / ln (4)
    = 2.7725887222 / 1.3862943611 = 2
  4. b = 5
    Log a (x) = log 5 (x) / log 5 (a)
    Log 4 (16) = log 5 (16) / log 5 (4)
    = 1.7227062322 / 0.8613531161 = 2
  5. b = 8
    Log a (x) = log 8 (x) / log 8 (a)
    Log 4 (16) = log 8 (16) / log 8 (4)
    = 1.3333333333 / 0.6666666666 = 2

Change of Base Rule in Logarithms

Our examples demonstrate that you can evaluate a non-standard-base logarithm log a (x) by converting it to a standard-base logarithm in fraction form log b (x) / log b (a) .

The nominator log b (x) is the standard-base logarithm of the non-standard-base logarithm’s exponent.

The denominator log b (a) is the standard-base logarithm of non-standard-base logarithm’s base.

How to Change the Base of a Log

  1. Decide on the Standard Base

    Decide which standard base you are going to use

  2. Rewrite the Non-standard Base Logarithm

    Rewrite the non-standard base logarithm as a fraction of a standard base logarithm

Frequently Asked Questions

lick on the question which is of interest to you to see the collapsible content answer.

How do you use the change of base formula?

Use the formula to evaluate a log in a base other than e or 10.

What is the change of base process?

It means either changing a value from one numeral system to another, for example from base 10 to base 2, or it refers to the procedure of changing the base of a logarithm.

How to find the base of a logarithmic function?

The base of a logarithmic function can be found right after the word “log”. It is usually written using subscript such that it is slightly below and usually lower than the normal line of type. If no base is explicitly stated, the base is 10, and ln has e as it’s base.

What is the change of base property?

It is a property that allows you to rewrite any log with another base.

Can you change the base of a log?

You can change the base of every log by applying the change of base rule.

How do you change the base of a log on a calculator?

Most calculators come with both, a ln as well as a log button. To enter the log with a different base, you will first need to apply the formula log a (x) = log b (x) / log b (a).

What does change of base formula create?

By using this formula, you can change the logarithm to any base you want.

How does the change of base rule work?

It works by expressing a logarithm in terms of given base in another base.


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