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

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- Oct 5th 2006, 09:55 AMsarahhLogarithm problem
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 - Oct 5th 2006, 10:15 AMtopsquark
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 - Oct 5th 2006, 02:44 PMQuick
- Oct 5th 2006, 02:53 PMtopsquark
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. :) ) - Oct 5th 2006, 03:41 PMdan
good explanation topsquark. that even sence to me!!!

dan - Oct 5th 2006, 05:09 PMThePerfectHacker
- Oct 5th 2006, 10:55 PMtopsquark
- Oct 5th 2006, 10:57 PMearboth
- Oct 5th 2006, 11:08 PMtopsquark
- Oct 6th 2006, 09:22 AMThePerfectHacker