Hi everyone,
Need a lot of help on this:
Find all real numbers x, if any, that satisfy the eq:
(log x^2)^2 = log x^4
NOTE: each log has base 10
I'm assuming the LHS is [log(x^2)]^2 and the RHS is log(x^4)?
Then
[2 log(x)]^2 = 4 log(x)
Set y = log(x)
[2y]^2 = 4y
4y^2 = 4y
y^2 = y
y^2 - y = 0
y(y - 1) = 0
So y = 0 or y = 1.
Thus log(x) = 0 or log(x) = 1
log(x) = 0
10^[log(x)] = 10^0
x = 1
and
log(x) = 1
10^[log(x)] = 10^1
x = 10.
Thus x = 1 or x = 10.
-Dan
The function
y = a^x
has an inverse defined as y = log(a) x. In English this reads the "log to the base a of x." (The "a" is subscripted.)
So we know that
a^[log(a) x] = x
and
[log(a) a^x] = x
Typically (though not universally, there's a thread on the forum somewhere on this) "log to the base 10" is usually shortened to the abreviation "log" (with no base mentioned) and "ln" is log(e) where e = 2.817...
-Dan
For general consumption, I fixed an error in my earlier post. (That's what I thought Quick was going to have posted about. )