How can one change the base of logarithms?

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.