SOLUTION: solve the solution set for x: 3^(2x+1) - 5*2^(2x+1) = 6^x

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Question 644781: solve the solution set for x:
3^(2x+1) - 5*2^(2x+1) = 6^x
Answer by Edwin McCravy(20054) About Me  (Show Source):
You can put this solution on YOUR website!

This requires these two principles of exponents

(1)    a%5E%28b%2Bc%29 = a%5Eb%2Aa%5Ec  <--The reversal of adding exponents

(2)    %28a%5Eb%29%2F%28c%5Eb%29 = %28a%2Fc%29%5Eb <--The reversal of the power of a fraction 


3%5E%282x%2B1%29 - 5%2A2%5E%282x%2B1%29 = 6%5Ex

Divide through by 2%5E%282x%2B1%29

3%5E%282x%2B1%29%2F2%5E%282x%2B1%29 - 5 = 6%5Ex%2F%282%5E%282x%2B1%29%29

Use principle (2) on the left side
Write 6 as 2*3 on the right

%283%2F2%29%5E%282x%2B1%29 - 5 = %282%2A3%29%5Ex%2F%282%5E%282x%2B1%29%29

Use principle (1) on the right side

%283%2F2%29%5E%282x%2B1%29 - 5 = %282%5Ex%2A3%5Ex%29%2F%282%5E%282x%2B1%29%29

Use principle (1) on the left
Subtract exponents of 2 on the right

%283%2F2%29%5E%282x%29%2A%283%2F2%29%5E1%29 - 5 = 3%5Ex%2F%282%5E%282x%2B1-x%29%29

Simplify

%283%2F2%29%5E%282x%29%2A%283%2F2%29%29 - 5 = 3%5Ex%2F%282%5E%28x%2B1%29%29

Swap factors on the left, use principle (1) on the right denominator

%283%2F2%29%283%2F2%29%5E%282x%29%29 - 5 = 3%5Ex%2F%282%5Ex%2A2%5E1%29%29

Remove the 1 exponent

%283%2F2%29%283%2F2%29%5E%282x%29%29 - 5 = 3%5Ex%2F%282%5Ex%2A2%29%29

Multiply through by 2

2%2A%283%2F2%29%283%2F2%29%5E%282x%29%29 - 2·5 = 2%2Aexpr%283%5Ex%2F%282%5Ex%2A2%29%29%29

cross%282%29%2A%283%2Fcross%282%29%29%283%2F2%29%5E%282x%29%29 - 10 = cross%282%29%2Aexpr%283%5Ex%2F%282%5Ex%2Across%282%29%29%29%29

3%283%2F2%29%5E%282x%29%29 - 10 = 3%5Ex%2F2%5Ex

3%283%2F2%29%5E%282x%29%29 - 10 = %283%2F2%29%5Ex

3%283%2F2%29%5E%282x%29%29 - %283%2F2%29%5Ex - 10 = 0

let u = %283%2F2%29%5Ex

Then uČ = %28%283%2F2%29%5Ex%29%5E2 = %283%2F2%29%5E%282x%29%29

Then we have

3uČ - u - 10 = 0

Factoring,

(u - 2)(3u + 5) = 0

Use the zero-factor principle:

u - 2 = 0;  3u + 5 = 0
    u = 2       3u = -5
                 u = -5%2F3

Use u = 2

Since u = %283%2F2%29%5Ex

%283%2F2%29%5Ex = 2

Take logs of both sides:

log%28%28%283%2F2%29%5Ex%29%29 = log%28%282%29%29

Use the principle log%28%28b%5Ea%29%29=a%2Alog%28%28b%29%29

x%2Alog%28%283%2F2%29%29 = log%28%282%29%29

Divide both sides by log%28%283%2F2%29%29

x = log%28%282%29%29%2Flog%28%283%2F2%29%29 = 1.709511291

That's the only solution because if we try 

u = -5%2F3

%283%2F2%29%5Ex = -5%2F3

That cannot produce an additional solution because because 3%2F2 is
positive and a positive number raised to any power, positive or negative,
gives a positive answer. 

Edwin