Power rule of Logarithms
Formula
$\log_{b}{\big(x^{\displaystyle n}\big)} \,=\, n \times \log_{b}{(x)}$
What is the power rule of logarithms?
The logarithm of a number raised to a power is equal to the power multiplied by the logarithm of that number.
This is the definition of the power law of logarithms and one of the most important laws of logarithms. In mathematics, it is used to convert exponents into coefficients and is widely used for simplifying logarithms.
In arithmetic, raising a number to a power means multiplying the number by itself as many times as indicated by the exponent. However, the logarithm of a number raised to a power cannot be evaluated in the same way.
Example
In arithmetic, $5^2$ $\,=\,$ $5 \times 5$ $\,=\,$ $25$
But, $\log{(5^2)}$ $\,\ne\,$ $\log{(5)}$ $\times$ $\log{(5)}$
To evaluate the logarithm of a number raised to a power, we use a special logarithmic identity called the power law of logarithms.
Examples of Power rule of logarithms
Now, let’s understand the logarithm power rule through a simple and clear example.
Example
Evaluate $\log_{4}{(4^3)}$
The argument of the logarithm is written in exponential form, but since it represents a numerical value, it can be evaluated easily by multiplying $4$ by itself repeatedly, as determined by the exponent.
$\,=\,\,$ $\log_{4}{(4 \times 4 \times 4)}$
$\,=\,\,$ $3$
As per the definition of the logarithm, the logarithm of a number $4$ raised to the power of $3$ to base $4$ is $3$.
Now, let’s find the logarithm of the numbers $4$ to base $4$.
$\log_{4}{(4)}$ $\,=\,$ $1$
Finally, multiply the logarithm of number $4$ to base $4$ by the exponent to find their product mathematically.
$\,=\,\,$ $3$ $\times$ $\log_{4}{(4)}$
$\,=\,\,$ $3$ $\times$ $1$
$\,=\,\,$ $3$
Now, compare both results to understand how the power rule for logarithms works.
$\,\,\,\,\therefore\,\,\,\,$ $\log_{4}{(4^3)}$ $\,=\,$ $3$ $\times$ $\log_{4}{(4)}$
The above numerical example explains why the logarithm of a number raised to a power is equal to the power multiplied by the logarithm of that number. This logarithmic property is known as the power rule in logarithms. The following examples further illustrate this rule for better understanding.
- $\log_{6}{(10^2)}$ $\,=\,$ $2\log_{6}{(10)}$
- $\log_{3}{(x^5)}$ $\,=\,$ $5\log_{3}{(x)}$
- $\log_{e}{(\sqrt{y})}$ $\,=\,$ $\dfrac{1}{2}\log_{e}{(y)}$
How to Write the Log Power Rule
The power property of logarithms is expressed mathematically as follows.
$\log_{b}{\big(x^{\displaystyle n}\big)} \,=\, n \times \log_{b}{(x)}$
Domain Conditions
It is one of the most useful logarithmic identities in mathematics, but it is important to understand the conditions under which the logarithmic power rule applies.
- $b$ denotes a quantity ($b > 0$ and $b \ne 1$) and the base of logarithm.
- $x$ represents a quantity and its value always positive. ($x > 0$)
- The value of $x$ raised to the power $n$ should always positive. ($x^n > 0$)
This explains how the logarithm of a number raised to a power converts the exponent into a coefficient multiplied by the logarithm of that number.
Derivation of the Log Power Law
Let’s learn how to prove that the logarithm of a number raised to a power is equal to the power multiplied by the logarithm of that number.
You have learned what the power law of logarithms is, explored its proof, and solved several examples to understand how this logarithmic formula works in mathematics.
Properties of Logarithms
Learn similar logarithmic properties that make simplifying expressions and solving problems easier.
$\log_{b}{(x.y)}$ $\,=\,$ $\log_{b}{(x)}$ $+$ $\log_{b}{(y)}$
$\log_{b}{\Big(\dfrac{x}{y}\Big)}$ $\,=\,$ $\log_{b}{(x)}$ $-$ $\log_{b}{(y)}$
$\log_{b}{(x)}$ $\,=\,$ $\dfrac{\log_{c}{(x)}}{\log_{c}{(b)}}$