SOLUTION: How do you solve this step by step? 3^(x-3)= (1/9)^-x

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Question 813878: How do you solve this step by step? 3^(x-3)= (1/9)^-x
Found 3 solutions by Alan3354, josmiceli, jsmallt9:
Answer by Alan3354(69443) About Me  (Show Source):
You can put this solution on YOUR website!
3^(x-3)= (1/9)^-x
3^(x-3)= 9^x
3^(x-3)= 3^(2x)
x-3 = 2x
x = -3

Answer by josmiceli(19441) About Me  (Show Source):
You can put this solution on YOUR website!
+3%5E%28x-3%29+=+%281%2F9%29%5E%28-x%29+
Note that +1%2F9+=+9%5E%28-1%29+, and
+9%5E%28-1%29+=+%28+3%5E2+%29%5E%28-1%29+
+%28+3%5E2+%29%5E%28-1%29+=+3%5E%28-2%29+
---------------------
Now I have
+3%5E%28x-3%29+=+%28+3%5E%28-2%29+%29%5E%28-x%29+
+3%5E%28+x-3+%29+=+3%5E%282x%29+
+x-3+=+2x+
+x+=+-3+
---------
check:
+3%5E%28x-3%29+=+%281%2F9%29%5E%28-x%29+
+3%5E%28-3-3%29+=+%281%2F9%29%5E%28-%28-3%29+%29+
+3%5E%28-6%29+=+%281%2F9%29%5E3+
+3%5E%28-6%29+=+%283%5E%28-2%29%29%5E3+
+3%5E%28-6%29+=+3%5E%28-6%29+
OK

Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
3%5E%28x-3%29=+%281%2F9%29%5E%28-x%29
When the variable is in an exponent there are generally two ways to solve it:
  • With logarithms. This can always be done.
  • If the equation can be rewritten two equal exponentials (a base and an exponent) with equal bases, then the equation can be solved much more quickly (and without decimal approximations).
Your equation already has two equal exponentials. But the bases, 3 and 1/9, are not the same. Unless we can rewrite these with the same bases we will have to use logarithms.

Fortunately 3 and 1/9 are both powers of 3. So we can rewrite your equation as:
3%5E%28x-3%29=+%283%5E%28-2%29%29%5E%28-x%29
Using the rule for powers of a power, i.e. multiply the exponents, we get:
3%5E%28x-3%29=+3%5E%282x%29
We now have the equation in the desired form. The next step is based on some simple logic. The equation says that two powers of 3 are equal. The only way this can be true is if the exponents themselves are equal, too. So:
x - 3 = 2x
This is easy to solve. Subtracting x from each side we get:
-3 = x