Question 74624
<pre><font size = 4><b>
: Solve: 
(2^x)-(3^x)=0 
I tried to bring it to base 2 but it didn't work. Please show us steps. 
Thank you. 

2<sup>x</sup> - 3<sup>x</sup> = 0

     2<sup>x</sup> = 3<sup>x</sup>

There are two log bases on your
calculator, "log" which is the 
logarithm base 10, and "ln" which
is the natural logarithm base "e".
You can use either one:

If we use "log", we take the log     |    If we use "ln", we take the ln
of both sides:                       |    of both sides:

  log(2<sup>x</sup>) = log(3<sup>x</sup>)                          ln(2<sup>x</sup>) = ln(3<sup>x</sup>)
                                     |
Now we use the rule of logs to       |    Now we use the rule of lns to
rewrite both sides:                  |   rewrite both sides:
                                     |
  log(A<sup>B</sup>) = B·log(A)                        ln(A<sup>B</sup>) = B·ln(A)
                                     |
 x·log(2) = x·log(3)                 |     x·ln(2) = x·ln(3)
                                     |
Get 0 on the right                   |   Get 0 on the right
                                     |
 x·log(2) - x·log(3) = 0             |    x·ln(2) - x·ln(3) = 0
                                     |
Factor out x on the left:            |   Factor out x on the left:     
                                     |
 x·[log(2) - log(3)] = 0             |    x·[ln(2) - ln(3)] = 0
                                     |
Using the calculator:                |   Using the calculator:
                                     |
     x·{.301 - .477) = 0             |    x·(.693 - 1.099) = 0
                                     |
           x·(-.176) = 0             |           x·(-.406) = 0
                                     |
              -.176x = 0             |              -.406x = 0
                                     |
Divide both sides by -.176           |  Divide both sides by -.406
                                     |
                   x = 0             |                   x = 0
                                     |
                    Either way you get x = 0 

Checking:
             2<sup>x</sup> - 3<sup>x</sup> = 0
             2<sup>0</sup> - 3<sup>0</sup> = 0
               1 - 1 = 0
                   0 = 0

Edwin</pre>