Logarithm question

Question

Log²(x-2)=4

i got the solution correct, it’s x=102, x=201/100 but I need the domain too and I got x>0 which is wrong. Please help!

Answer 1

take square roots of both sides to get

Log(x-2) = 2 or -2

put both sides as powers of 10

x-2 = 10^2 = 100

x = 102 or

x-2 = 10^-2 = 1/100

x = 1/100+2 = 2.01

You got that right. sort of. See last comments below for alternative solution.

domain for a log is you can't take the log of a negative number, so you're tempted to say x>o

but this is log of (x-2) so x-2>0 which means x>2 as the domain

You interpreted log as log to the base 10. That's common, but higher math often treats log as ln, the natural log, (they like natural logs over logs to base 10 in higher mathematics), so if you do that then

ln(x-2) = 2 and

x-2 = +e^2

x = 2 + e^2 = about 2 + or - 7.5 = -5.5 or +9.5

still with domain x>2