Question

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

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

Answer 1

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.