# How can one change the base of logarithms?

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### 4 Answers

First let us start with what logarithm is.

log (a) b = c

=> b = a^c

Take the log to the base k of both the sides

=> log (k) b = log (k) a^c

use the power rule of logarithms that states log a^b = b*log a

=> log (k) b = c log (k) a

divide both sides by log (k) a

=> log (k) b / log (k) a = c (log (k) a/ log (k) a)

=> c = log (k) b/ log (k) a

Therefore to change the base of a logarithm from a to k, we change the base of the original logarithm to k, and divide it by log (k) a.

You can use the following logarithm properties to change the base.

log(a) b = log(c) b / log(c) a

For example:

log(3) 12 = log(4) 12 / log(4) 3

OR

log5 (16) = log(4) 16 / log(4) 5.

And you also can change to the base 10 as well.

For example:

log3 x = log x / log 3 ( The base here is 10).

Hope that helps.

We know that if a^x = y, then log (a) y = x.

Or if log (a) y = x, then a^x = y.

Now how to change the base log (a) y = x to the base b:

a^x = y.

Let b^c = a which implies log(b) a = c.

Then a^x = y implies ( b^c)^x = y.

b^cx = y implies log (b) y = xc.

=>lg(b) y = xlog (b)a.

=> log(b) y / log (b) a = x.

Therefore if log(a) y = x, then log(b) y / log (b) a = x.

Example and verification:

We know log (100) 100 00 00 = log(100) 100^3 = 3.

log (100) 1000000 = {log (10) 1000000}/log(10) 100 = {log(10) 10^6}/ log (10) 10^2 = 6/2 = 3.

You can recall the phrase: the logarithm in the new base, of the number is the logarithm in the old base, of the number, multiplied by the logarithm in the new base, of the old base.

For instance, we have the log 4 8 and we'll have to change the base 5 into the base 2:

log 2 8 = log 4 8*log 2 4

log 2 (2^3) = log 4 8* log 2 (2^2)

3 log 2 2 = (log 4 8)*2(log 2 2)

But log 2 2 = 1

3 = 2(log 4 8)

**(log 4 8) = 3/2**