Table of Contents

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)

## Examples

- b = 2 Log
_{a}(x) = log_{2}(x) / log_{2}(a) Log_{4}(16) = log_{2}(16) / log_{2}(4) = 4 / 2 = 2 - b = 10 Log
_{a}(x) = log (x) / log (a) Log_{4}(16) = log (16) / log (4) = 1.2041199826 / 0.6020599913 = 2 - b = e Log
_{a}(x) = ln (x) / ln (a) Log_{4}(16) = ln(16) / ln (4) = 2.7725887222 / 1.3862943611 = 2 - 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 - 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

**Decide on the Standard Base**Decide which standard base you are going to use

**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?

### What is the change of base process?

### How to find the base of a logarithmic function?

### What is the change of base property?

### Can you change the base of a log?

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

_{a}(x) = log

_{b}(x) / log

_{b}(a).

### What does change of base formula create?

### How does the change of base rule work?

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