Why does the change of base rule for logarithms work for any value of x?

logb(a)=(logx(a))/(logx(b)

1 Answer
Jun 18, 2018

To see why the change of base formula works for any x, we need to see how it is derived.

Explanation:

Let log_b(a)=r.
By the definition of a logarithm, a=b^r.
Now we can take the base x logarithm of both sides of the equation to end up with log_x(a)=log_x(b^r).
Simplifying:
log_x(a)=rlog_x(b)
log_x(a)/log_x(b)=r
Substituting r=log_b(a), we get
log_a(b)=log_x(a)/log_x(b).
Therefore, the change of base formula works for any number x as long as log_x is defined.