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

1 Answer
Sep 11, 2015

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