SOLUTION: 3(9)^(x-1)=(81)^(2x+1)

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Question 377487: 3(9)^(x-1)=(81)^(2x+1)
Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
3%289%29%5E%28x-1%29=%2881%29%5E%282x%2B1%29
Since the 3, the 9 and the 81 are all powers of 3, we can rewrite each side of this equation as powers of 3. This is probably the easiest way to solve this equation.
3%283%5E2%29%5E%28x-1%29=%283%5E4%29%5E%282x%2B1%29
On each side we have a power to a power. The exponent rule for this is to multiply the exponents. This gives us:
3%283%5E%282x-2%29%29=3%5E%288x%2B4%29
On the left side we have 3 (or 3%5E1) times a power of three. The exponent rule for this is to add the exponents:
3%5E%281%2B2x-2%29=3%5E%288x%2B4%29
which simplifies to:
3%5E%282x-1%29=3%5E%288x%2B4%29
We now have both sides as powers of 3. The only way these powers of 3 can be equal if if the exponents are equal. So:
2x-1 = 8x+4
This is an easy equation to solve. Subtract 2x from each side:
-1 = 6x + 4
Subtract 4 from each side:
-5 = 6x
Divide both sides by 6:
-5%2F6+=+x
And we're done.

This problem can also be done with logarithms. And if the 3, the 9 and the 81 had not all been powers of the same number we would have to use logarithms. We could use base 3 logarithms:
log%283%2C+%283%289%29%5E%28x-1%29%29%29=log%283%2C+%2881%29%5E%282x%2B1%29%29

log%283%2C+%283%29%29+%2B+%28x-1%29log%283%2C+%289%29%29=+%282x%2B1%29log%283%2C+%2881%29%29




Since 3+=+3%5E1, 9+=+3%5E2 and 81+=+3%5E4 the base 3 logs of each are 1, 2 and 4, respectively. Substituting these in to the equation we get:
%281+-+2+-+4%29%2F%282%2A4+-+2%29+=+x
%28-5%29%2F6+=+x

We could also use base 10 logarithms:
log%28%283%289%29%5E%28x-1%29%29%29=log%28%2881%5E%282x%2B1%29%29%29
log%28%283%29%29+%2B+log%28%28%289%29%5E%28x-1%29%29%29=log%28%2881%5E%282x%2B1%29%29%29
log%28%283%29%29+%2B+%28x-1%29log%28%289%29%29=+%282x%2B1%29log%28%2881%29%29




We could now get out our calculators and find a decimal approximation for x. This should work out to something very close to the decimal form of -5/6: -0.8333333....