Question

prove that log a^m=m log a...?

Answer 1

log a^m = m log a

First of all, you need to know that to remove the power from a number in a log expression, always move the power to the front of the log.

Since m is the power in our log expression, move it to the front of the log to get:

m log a

Therefore, log a^m = m log a. (proven)

Answer 2

To prove :- log(a)^m = m log(a)

Let, log(a)^m = x

Hence, $10^{x} = a^{m}$

(Since, it is a natural log of base 10)

=> $10^{\frac{x}{m}} = a$

Now put the value of a in the logarithm value

=> log(a) = $log(10^{\frac{x}{m}})$

Now, $log(10^{\frac{x}{m}}) = \frac{x}{m}$

So, $\frac{x}{m} = log(a)$

Cross multiply,

Hence, x = m log(a)

But we have taken that x = $log(a)^{m}$

=> $log(a)^{m} = m log(a)$

Hence Proved