Question 25373
<pre>If 5^x=3 and 9^y=125, find the value of xy.
<b><font size = 4>
For the first equation

     5<sup>x</sup> = 3

Take the log of both sides

log(5<sup>x</sup>) = log(3)

Use the rule of logs on LHS:  logA<sup>B</sup> = B·logA

x·log5 = log3

Divide both sides by log5

    log3   
x = ———— 
    log5  

====

For the second equation

      9<sup>y</sup> = 125

Take the log of both sides

 log(9<sup>y</sup>) = log125

As before, use the rule of logs on LHS:  logA<sup>B</sup> = 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 5<sup>3</sup> and 9 as 3<sup>2</sup>

log3   log5<sup>3</sup>
———— · ——————
log5   log3<sup>2</sup>

Again we use the rule  logA<sup>B</sup> = B·logA 

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

Cancel the log3's

 1
<s>log3</s>   3log5
———— · ——————
log5   2<s>log3</s>
         1    

Cancel the log5's

 1        1
<s>log3</s>   3<s>log5</s>
———— · ——————
<s>log5</s>   2<s>log3</s>
 1        1    

All that's left is

 3
———
 2

Edwin
AnlytcPhil@aol.com</pre>