Base Changing Breakdown

In the following argument in which step has a mistake been made:

let logₘn = a

changing the base to 10 gives

1. a = $\frac{log₁₀n}{log₁₀m}$

2. a = log₁₀( $\frac{n}{m}$ )

3. 10ᵃ = $\frac{n}{m}$

however changing the base to 11 instead of 10 gives

11ᵃ = $\frac{n}{m}$

thus 10ᵃ = 11ᵃ

• or 10 = 11 !