Question 38907
Hello, 
I need help with a few problems i cant seem to work out for homework. 
1:Solve for x: 9^x=3
2:Solve for x: 2^9x^-4=32
3:Solve for x: log10^7=log10x-log10^2 
<pre><font size = 4><b>
9<sup>x</sup> = 3

We can relate the two numbers 3 and 9 by the equation

9 = 3<sup>2</sup>

So we substitute (32) for 9

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

Remove the parentheses by multiplying outer 
exponent x by inner exponent 2

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

Now write the 3 on the right as 31

3<sup>2x</sup> = 3<sup>1</sup>

Now use the fact that if two exponential expressions are equal
and the common base is positive and not 1, then the exponents are
equal, so we equate the exponents of 3 on both sides:

2x = 1

 x = 1/2

==========================================================

2^9x^-4=32

Big problem here! I can't tell if that means

(A)   2<sup>9</sup>x<sup>-4</sup> = 32

or

(B)   2<sup>9x<sup>-4</sup></sup> = 32

or

(C)   (2<sup>9x</sup>)<sup>-4</sup> = 32

If it means (A)

2<sup>9</sup>x<sup>-4</sup> = 32

Substitute 2<sup>5</sup> for 32

2<sup>9</sup>x<sup>-4</sup> = 2<sup>5</sup>

Divide both sides by 2<sup>9</sup>

    x<sup>-4</sup> = 2<sup>5</sup>/2<sup>9</sup>

    x<sup>-4</sup> = 2<sup>-4</sup>

Raise both sides to the -1/4 power

    (x<sup>-4</sup>)<sup>-1/4</sup> = (2<sup>-4</sup>)<sup>-1/4</sup>
 
        x<sup>1</sup> = 2<sup>1</sup>        

         x = 2                       

====================================================

If it means (B)

(B)   2<sup>9x<sup>-4</sup></sup> = 32
    
Substitute 2<sup>5</sup> for 32

  2<sup>9x<sup>-4</sup></sup> = 2<sup>5</sup>

Now use the fact that if two exponential expressions are equal
and the common base is positive and not 1, then the exponents are
equal, so we equate the exponents of 2 on both sides:

    9x<sup>-4</sup> = 5

Divide both sides by 9

     x<sup>-4</sup> = 5/9   

Raise both sides to the -1/4 power

    (x-4)<sup>-1/4</sup> = (5/9)<sup>-1/4</sup>

     x<sup>1</sup> = (9/5)<sup>1/4</sup>

      x = (9/5)<sup>1/4</sup>

      x = 1.158292185 approximately

===============================================

If it means (C)

(2<sup>9x</sup>)<sup>-4</sup> = 32

Remove the parentheses on the left by 
multiplying the outer exponent -4 by the
inner exponent 9x. On the right, substitute
2<sup>5</sup> for 32

2<sup>-36x</sup> = 2<sup>5</sup>

Now use the fact that if two exponential expressions are equal
and the common base is positive and not 1, then the exponents are
equal, so we equate the exponents of 2 on both sides:

-36x = 5

  x = -5/36

==========================================================

log(10<sup>7</sup>) = log(10x) - log(10<sup>2</sup>)

Use fact that log(10<sup>N</sup>) = N

     7 = log(10x) - 2

     9 = log(10x)

Use the fact that log(AB) = log(A) + log(B)

     9 = log(10) + log(x)

Use the fact that log(10) = log(10<sup>1</sup>) = 1

     9 = 1 + log(x)

     8 = log(x)

Use the fact that equation A = log(B) can be written
as equation B = 10<sup>A</sup> 

    x = 10<sup>8</sup>

    x = 100,000,000    

Edwin
AnlytcPhil@aol.com</pre></font>