Question

What is x if `log_4(8x ) - 2 = log_4 (x-1)`?

Answer 1

`x=2`

Explanation:

We would like to have an expression like

`log_4(a)=log_4(b)`, because if we had it, we could finish easily, observing that the equation would the solved if and only if `a=b`. So, let's do some manipulations:

1. First of all, note that `4^2=16`, so `2=log_4(16)`.

The equation then rewrites as

`log_4(8x)-log_4(16)=log_4(x-1)`

But we're still not happy, because we have the difference of two logarithms in the left member, and we want a unique one. So we use

1. `log(a)-log(b)=log(a/b)`

So, the equation becomes

`log_4(8x/16)=log_4(x-1)`

Which is of course

`log_4(x/2)=log_4(x-1)`

Now we are in the desired form: since the logarithm is injective, if `log_4(a)=log_4(b)`, then necessarily `a=b`. In our case,

`log_4(x/2)=log_4(x-1) iff x/2 = x-1`

Which is easily solve into `x=2x-2`, which yields `x=2`