Question 819620: Solve each exponential equation for all the real solutions
9^2x+1=27^x+5
Answer by jsmallt9(3758)   (Show Source):
You can put this solution on YOUR website! What you posted meant:
But I'm guessing that the exponents are supposed to be (2x+1) and (x+5). If I am wrong then re-post your question (since I will be answering the wrong problem. If I am right, then in the future please put exponents (numerators, denominators, radicands, function arguments, etc.) in parentheses, especially if they are more than just a positive integer or variable.
With variables in exponents we could use logarithms to solve this equation. But there is a faster, easier, more accurate way of solving the equation if it is possible to rewrite it as two powers of the same base being equal. So if we could- Rewrite the right side so it is a power of 9. Or...
- Rewrite the left side so it is a power of 27. Or...
- Rewrite both sides as some power of some third number.
... then we could use this faster method.
With a little effort we should be able to figure out that while 9 and 27 are not well=known powers of each other they are both powers of 3. So we can rewrite both sides as powers of 3:
Use the power of a power rule (i.e. multiply the exponents) we get:
Now that the equation is in the desired form, the next step is based on a simple fact: The only way two powers of 3 can be equal is if the exponents are equal. So:
This is a very simple equation to solve. Subtracting 3x:
Subtracting 2:
P.S. If we had used logarithms, not only would it have been harder but we might have gotten an answer like 13.000001 or 12.9999997 (very close but not quite correct) because of the decimal approximations we would have used.
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