9x = 3

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

9 = 32

So we substitute (32) for 9

(32)x = 3

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

32x = 3

Now write the 3 on the right as 31

32x = 31

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)   29x-4 = 32

or

(B)   29x-4 = 32

or

(C)   (29x)-4 = 32

If it means (A)

29x-4 = 32

Substitute 25 for 32

29x-4 = 25

Divide both sides by 29

    x-4 = 25/29

    x-4 = 2-4

Raise both sides to the -1/4 power

    (x-4)-1/4 = (2-4)-1/4
 
        x1 = 21        

         x = 2                       

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

If it means (B)

(B)   29x-4 = 32
    
Substitute 25 for 32

  29x-4 = 25

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-4 = 5

Divide both sides by 9

     x-4 = 5/9   

Raise both sides to the -1/4 power

    (x-4)-1/4 = (5/9)-1/4

     x1 = (9/5)1/4

      x = (9/5)1/4

      x = 1.158292185 approximately

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

If it means (C)

(29x)-4 = 32

Remove the parentheses on the left by 
multiplying the outer exponent -4 by the
inner exponent 9x. On the right, substitute
25 for 32

2-36x = 25

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(107) = log(10x) - log(102)

Use fact that log(10N) = 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(101) = 1

     9 = 1 + log(x)

     8 = log(x)

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

    x = 108

    x = 100,000,000    

Edwin
AnlytcPhil@aol.com