Find the solution s of the logarithmic equation log x log x

Find the solution (s) of the logarithmic equation log x + log (x - 20) = log (2x) correct to four decimal places. If there is more than one solution write them separated by commas.

Solution

Using properties of logarithm we can combine the left hand side into a single term:

ln(x(x-20)) = ln(2x)

Now we can get rid of log

x (x-20) = 2x

x²-20x = 2x

x² = 22x

x (x-22) = 0

x = 0 and x = 22

But,

0 cannot be inputted to log. Therefore only valid answer is 22.