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

Question 377487: 3(9)^(x-1)=(81)^(2x+1)
Answer by jsmallt9(3758) (Show Source): You can put this solution on YOUR website!

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.

On each side we have a power to a power. The exponent rule for this is to multiply the exponents. This gives us:

On the left side we have 3 (or ) times a power of three. The exponent rule for this is to add the exponents:

which simplifies to:

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:

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:

Since , and the base 3 logs of each are 1, 2 and 4, respectively. Substituting these in to the equation we get:

We could also use base 10 logarithms:

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....
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