Logarith question.

**DannyMath** · September 20th 2010, 06:45 PM

I have to solve:

(log(x^2) + log(x^4))/log(90x) = 3

I've been on it for 25 minutes and have gotten as far as:

log(x)/log(90x) = 1/2

And I'm not sure if I'm even going the right path lol, just trying to play around with properties of logs and not sure how to maneuver this. Thanks for any help

**CaptainBlack** · September 20th 2010, 08:16 PM

Originally Posted by DannyMath

I have to solve:

(log(x^2) + log(x^4))/log(90x) = 3

I've been on it for 25 minutes and have gotten as far as:

log(x)/log(90x) = 1/2

And I'm not sure if I'm even going the right path lol, just trying to play around with properties of logs and not sure how to maneuver this. Thanks for any help

Applying the laws of logarithms to reduce the left hand sid to an exoression in $\log(x)$ :

$\dfrac{\log(x^2)+\log(x^4)}{\log(90x)}=\dfrac{2\lo g(x)+4\log(x)}{\log(90)+\log(x)}$

and you should be able to finish from there.

CB

**Soroban** · September 20th 2010, 08:27 PM

Hello, DannyMath!

What you did was correct . . .

$\text{Solve for }x\!:\;\;\dfrac{\log(x^2) + \log(x^4)}{\log(90x)} \:=\: 3$

Note that: . $x \ne 0$

We have: . $\log(x^2\cdot x^4) \;=\;3\log(90x) \quad\Rightarrow\quad \log(x^6) \:=\:3\log(90x)$

. . . . . . . . . . $6\log(x) \:=\:3\log(90x) \quad\Rightarrow\quad 2\log(x) \:=\:\log(90x)$

. . . . . . . . . . . $\log(x^2) \:=\:\log(90x) \quad\;\;\Rightarrow\qquad \quad\; x^2 \:=\:90x$

Then: . $x^2 - 90x \:=\:0 \quad\Rightarrow\quad x(x-90) \:=\:0$

Therefore: . $\rlap{//////}x \,=\,0,\;\;x\,=\,90$

**DannyMath** · September 20th 2010, 08:46 PM

Originally Posted by Soroban

Hello, DannyMath!

What you did was correct . . .

Note that: . $x \ne 0$

We have: . $\log(x^2\cdot x^4) \;=\;3\log(90x) \quad\Rightarrow\quad \log(x^6) \:=\:3\log(90x)$

. . . . . . . . . . $6\log(x) \:=\:3\log(90x) \quad\Rightarrow\quad 2\log(x) \:=\:\log(90x)$

. . . . . . . . . . . $\log(x^2) \:=\:\log(90x) \quad\;\;\Rightarrow\qquad \quad\; x^2 \:=\:90x$

Then: . $x^2 - 90x \:=\:0 \quad\Rightarrow\quad x(x-90) \:=\:0$

Therefore: . $\rlap{//////}x \,=\,0,\;\;x\,=\,90$

UGH I was getting so frustrated and you've made it so clear. Thank you! *sigh of relief*

**CaptainBlack** · September 21st 2010, 12:03 AM

Originally Posted by DannyMath

UGH I was getting so frustrated and you've made it so clear. Thank you! *sigh of relief*

I'm afraid he has actually made rather a meal of it.

Which I suppose is OK if you are going to spoon feed someone.

CB

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