Question

How do you solve `log 5 = (x+1) log 4`?

Answer 1

This would be same as solving a multi-step linear equation only the coefficients and constants are logarithmic numbers. Step by step working is shown below.

Explanation:

`log(5)=(x+1)log(4)`

Let us distribute `(x+1)log(4)`

`log(5) = xlog(4)+log(4)`

What we see here is not different from any multi-step linear equation .
Say if you had no log and the equation was `5=4x+4` you would have found it easy!

We shall use the same trick to solve it, but with the log present.

`log(5)=xlog(4)+log(4)`
First subtract `log(4)` on both the sides, this is done to isolate the `x`

`log(5)-log(4) = xlog(4)`

`log(5/4)=xlog(4)` Note : `log(P) - log(Q) = log(P/Q)`

Now we divide both sides by `log(4)` to isolate `x`

`log(5/4)/log(4) = x`

The exact solution is `x=log(5/4)/log(4)`

If you want an decimal approximation then you can use a calculator to find it.