Solve for x. 7^(x-2)=12^(x)

Question

Solve for x.      7^(x-2)=12^(x)

superhelp101 · Answer

@dumbcow  any help :))

dumbcow · Answer

use logs  
$$\ln a^n = n \ln a$$

superhelp101 · Answer

can you explain that ? :/

dumbcow · Answer

its a log property you will use to solve this 

whenever the exponent "x" is a variable, you can take the log and bring the exponent outside of the log

superhelp101 · Answer

oh okay, makes sense

superhelp101 · Answer

i am a bit stuck. Can you show me ?

dumbcow · Answer

$$\rightarrow (x-2) \ln 7 = x \ln 12$$

superhelp101 · Answer

would I then divide each side by ln12 to get x ?

dumbcow · Answer

no because there is an "x" on other side too

distribute, then put "x" terms on same side
factor out the "x" , then you divide

superhelp101 · Answer

how do i do that ?

dumbcow · Answer

just like how i described :)

do you know how to distribute?

superhelp101 · Answer

distribute what ?

superhelp101 · Answer

(x-2) ln7  ?

dumbcow · Answer

yes ... sorry got to go

superhelp101 · Answer

xln(7)-2ln(7)

superhelp101 · Answer

wait but what do I do next..... :((

superhelp101 · Answer

@mathmale can you plzz continue from where dumbcow left off, he had to go ?

mathmale · Answer

7^(x-2)=12^(x)   became  $$\rightarrow (x-2) \ln 7 = x \ln 12$$   (This is correct.  Any questions regarding what's happening here?)

superhelp101 · Answer

no

mathmale · Answer

Our job is to solve for x.  In other words, we want to isolate x on the left side of this equation and have everything else on the right side.

superhelp101 · Answer

okay

mathmale · Answer

Please multiply out the left side of $$\ (x-2) \ln 7 = x \ln 12$$

superhelp101 · Answer

xln(7)-21ln(7)=xln(12)

mathmale · Answer

Which of these terms contain a variable and which does/do not?

mathmale · Answer

Collect all with x in them on the left; collect all constants on the right.

superhelp101 · Answer

so I need to add 21ln(7) and subtract xlm(12) from both sides

mathmale · Answer

Yes, that's right.  Try it.

superhelp101 · Answer

ok so: xln(7)-xln(12)=21ln(7) But what happens next here ?

mathmale · Answer

Both terms on the left have x in them.  Factor out x and write the remaining factor inside parentheses.  Leave the right side alone.

superhelp101 · Answer

ok, that makes sense, but I do I factor the x ?

superhelp101 · Answer

@mathmale

mathmale · Answer

xln(7)-xln(12)=21ln(7)  becomes x (ln 7 - ln 12) = 21 ln 7.

superhelp101 · Answer

oh okay

mathmale · Answer

How would you now solve for x?

superhelp101 · Answer

do i use a calculator and solve the inside parenthesis first ?

mathmale · Answer

No.  Best to use a certain "rule of logs" to combine ln 7 - ln 12 into one log.

Of course, if you're supposed to find the answer as a decimal fraction, then go ahead and use your calculator.

superhelp101 · Answer

ok how would i combine, can you teach me it ?

mathmale · Answer

The rule of logs that applies here is log a - log b = log (a/b).
Therefore,                                         log 7 -log 12 =    ?

superhelp101 · Answer

log(7/12)

superhelp101 · Answer

ok I got it from there, thank youuuuuu...hhehe :)))) What are the benefits from getting the membership on os. Do you like it ?

mathmale · Answer

You're welcome!  I'm a retired teacher and love to continue working with students on their math through OpenStudy.  I'm also a Moderator and can thus help keep OS clean and focused.

superhelp101 · Answer

That's awesome! :) you're very kind! :)

superhelp101 · Answer

wait, but I try to solve x and I get -16.625 and when I plug it in it doesn't seem to work

superhelp101 · Answer

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