SOLUTION: If 5^x=3 and 9^y=125, find the value of xy.

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Question 25373: If 5^x=3 and 9^y=125, find the value of xy.
Answer by AnlytcPhil(1806) About Me  (Show Source):
You can put this solution on YOUR website!
If 5^x=3 and 9^y=125, find the value of xy.

For the first equation

     5x = 3

Take the log of both sides

log(5x) = log(3)

Use the rule of logs on LHS:  logAB = B·logA

x·log5 = log3

Divide both sides by log5

    log3   
x = ———— 
    log5  

====

For the second equation

      9y = 125

Take the log of both sides

 log(9y) = log125

As before, use the rule of logs on LHS:  logAB = B·logA

y·log9 = log125

Divide both sides by log9

    log125
y = ——————
     log9 

        log3   log125
So xy = ———— · ——————
        log5    log9


But we can simplify that

log3   log125
———— · ——————
log5    log9

write 125 as 53 and 9 as 32

log3   log53
———— · ——————
log5   log32

Again we use the rule  logAB = B·logA 

log3   3log5
———— · ——————
log5   2log3

Cancel the log3's

 1
log3   3log5
———— · ——————
log5   2log3
         1    

Cancel the log5's

 1        1
log3   3log5
———— · ——————
log5   2log3
 1        1    

All that's left is

 3
———
 2

Edwin
AnlytcPhil@aol.com