Question

How do you solve `log(x + 10) - log(x) = 2log(5)`?

Answer 1

`x = 5/12`

Explanation:

`log(x + 10) - logx = 2log5`

Our first step is to rewrite the equation using some laws of logarithms, specifically, `logA - logB = log (A/B)`. This law allows us to rewrite the left-hand side of the equation as

`log((x + 10)/x) = 2log5`.

Another law of logarithms, `A log B = log B^A`, allows us to rewrite the right-hand side equivalently as

`log((x + 10)/x) = log5^2`
`= log 25`

Now, you didn't specify a base for the `log` function here, so I will assume that `log` means base-2 logarithm. Still, whether the base is 2, 10, e, or whatever, it actually doesn't matter... the answer will be the same. You'll see why in a moment.

Raise 2 to both sides:

`2^log((x + 10)/x) = 2^log25`

The exponential and the logarithm are inverse functions, so the base-2 and the logarithms will cancel:

`(x + 10)/x = 25`

From here, we just need to use some simple algebra, multiplying both sides by `x`:

`x + 10 = 25x`

and then subtracting `x` from both sides:

`10 = 24x`

And then simplify to arrive at our final answer `x`:

`x = 5/12`