SOLUTION: 5^2x+3=(square root 5)^x+4

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Question 398979: 5^2x+3=(square root 5)^x+4
Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
Please use parentheses around multiple term exponents, numerators, denominators, arguments to functions, etc. Without them your equation reads as:
5%5E2x%2B3=sqrt%285%29%5Ex%2B4
and not what I am guessing it is supposed to be:
5%5E%282x%2B3%29=sqrt%285%29%5E%28x%2B4%29
Tutors will be more likely to help if the problem is clearly stated.

Assuming your equation is really:
5%5E%282x%2B3%29=sqrt%285%29%5E%28x%2B4%29
Solving equations where the variable is in one ore more exponents, like this equation, often involves use of logarithms. An exception to this is when both sides of the equation can be written as powers of the same number. Since 5 and sqrt%285%29 are both powers of 5, we will be able to write both sides as powers of 5.
And the rule for exponents when raising a power to a power is to multiply the exponents. So this gives us:
5%5E%282x%2B3%29=5%5E%28%281%2F2%29%2A%28x%2B4%29%29
which simplifies to:
5%5E%282x%2B3%29=5%5E%28%281%2F2%29x%2B2%29
We now have both sides of the equation as powers of 5. In order for these powers of 5 to be equal, the exponents must be equal. So:
2x%2B3=%281%2F2%29x%2B2
This equation is easily solved. First let's get rid of the fraction by multiplying both sides by 2:
2%282x%2B3%29=2%28%281%2F2%29x%2B2%29
which simplifies to:
4x + 6 = x + 4
Subtracting x from each side:
3x + 6 = 4
Subtracting 6 from each side:
3x = -2
Dividing both sides by 3:
x = -2/3