Question

How do you expand $\log {{N}^{3}}$?

Answer 1

We solve the given equation $\log {{N}^{3}}$ using the particular identity formula of logarithm like $\log {{x}^{a}}=a\log x$. The main step would be to eliminate the power value of the logarithm functions and keep it as a simple logarithm. we solve the linear multiplication with the help of basic binary operations.

Complete step by step solution:
We take the logarithmic identity for the given equation $\log {{N}^{3}}$ to find the solution for condensation.
For condensed form of logarithm, we apply power property, products of factors and logarithm of a power.
For our given equation we are only going to apply the power property.
We have $\log {{x}^{a}}=a\log x$. The power value of $a$ goes as a multiplication with $\log x$.
In case of logarithmic numbers having powers, we have to multiply the power in front of the logarithm to get the single logarithmic function.
Now we place the values of $a=3$ and $x=N$ in the equation of $\log {{x}^{a}}=a\log x$.
We get $\log {{N}^{3}}=3\log N$.
Therefore, the simplified form of $\log {{N}^{3}}$ is $3\log N$.

Note: To solve the logarithm we can also use the formula of $\log \left( abc \right)=\log a+\log b+\log c$. We replace the values and get
$\log {{N}^{3}}=\log \left( N\times N\times N \right)=\log N+\log N+\log N=3\log N$.
The formula $\log {{x}^{a}}=a\log x$ is a simplified form of $\log \left( abc \right)=\log a+\log b+\log c$.