SOLUTION: Hi. I've been trying to solve this for about three hours. I know the solution is 11.0834645967, but I can't figure out how to get it within the constraints of the problem: Show h

Algebra ->  Logarithm Solvers, Trainers and Word Problems -> SOLUTION: Hi. I've been trying to solve this for about three hours. I know the solution is 11.0834645967, but I can't figure out how to get it within the constraints of the problem: Show h      Log On


   

Question 136498: Hi. I've been trying to solve this for about three hours. I know the solution is 11.0834645967, but I can't figure out how to get it within the constraints of the problem: Show how to solve the following equation in terns of the common logarithm. Also give an approximation of the solution to the nearest hundreth of a unit. 3^(2x-3)=5^(x+2)
My work is
log(3^2x-3)=log(5^x+2)
(2x-3)log3=(x+2)log5
2x-3=(x+2)log5/log3
(2x-3)/(x+2)=log5/log3
(2x-3)/(x+2)=logbase3 5
and then I get stuck. And I might have gone right past the correct route and gotten more tangled. Any help would be most appreciated.
Answer by Edwin McCravy(20056) About Me  (Show Source):
You can put this solution on YOUR website!
Hi. I've been trying to solve this for about three hours. I know the solution is 11.0834645967, but I can't figure out how to get it within the constraints of the problem: Show how to solve the following equation in terns of the common logarithm. Also give an approximation of the solution to the nearest hundreth of a unit.
3%5E%282x-3%29=5%5E%28x%2B2%29
My work is
++log%28%28++3%5E%282x-3%29+%29%29+=+log%28%28++5%5E%28x%2B2%29++%29%29++
+%282x-3%29log%28%283%29%29+=+%28x%2B2%29log%28%285%29%29
2x-3=%28x%2B2%29%28log%28%285%29%29%2Flog%28%283%29%29%29

You did fine to here.  Then you went astray. Now here is what 
you should have done:

Get your calculator at this point and calculate %28log%28%285%29%29%2Flog%28%283%29%29%29 as

1.464973521.

and store it in the memory of your calculator, or write it down.
So you won't have to cary such a long awkward decimal through your
steps, just call it A.  In other words let A+=+%28log%28%285%29%29%2Flog%28%283%29%29%29
1.464973521

Now 

          2x-3=%28x%2B2%29%28log%28%285%29%29%2Flog%28%283%29%29%29

becomes simply

          2x-3=%28x%2B2%29A

or switch the A to the left of the parentheses:

          2x-3=A%28x%2B2%29

Distribute on the right:

          2x-3=Ax%2B2A

Get all the x terms on the left and other terms on the right:

          2x-Ax=3%2B2A

Factor out x on the left:

          x%282-A%29=3%2B2A

Divide both sides by %282-A%29

          %28x%282-A%29%29%2F%28%282-A%29%29=%283%2B2A%29%2F%28%282-A%29%29

Cancel the 2-A's

          x%2Across%28%282-A%29%29%2Fcross%28%282-A%29%29=%283%2B2A%29%2F%28%282-A%29%29

          x=%283%2B2A%29%2F%282-A%29

Now using the recall feature of your calculator to substitute
the value of A.

          x=%283%2B2%281.464973521%29%29%2F%282-1.464973521%29

Now calculate that with your calculator, and you'll get

          x=11.0834645967

Edwin