SOLUTION: Exponential and logarithmic equations and inequalities Solve: 3^x=1/27

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Question 712887: Exponential and logarithmic equations and inequalities
Solve: 3^x=1/27
Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
3%5Ex+=+1%2F27
When the variable is in an exponent like in this equation, logarithms are often used to solve the equation. But if the equation can be rewritten so that each side is a power of the same number there is a much shorter, easier way. It is worth the effort to see if this is possible.

So we will start by seeing if we can rewrite the equation so that each side is a power of the same number. The left side is already a power of 3. Can the right side be rewritten as a power of 3? It should take long to find that 27+=+3%5E3. And since 1%2Fa%5En+=+a%5E%28-n%29, 1%2F27+=+3%5E%28%28-3%29%29. So we can rewrite the equation as:
3%5Ex+=+3%5E%28%28-3%29%29
We now have the desired form.

With this form we have an equation that tells us that 3 to the x power is equal to 3 to the -3 power. The only way powers of 3 can be equal is if their exponents are equal. So:
x = -3
And we're done!!