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Thread: Intermediate Challenging Algebra
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Old 03-11-2015   #2
MAS1

 
Join Date: Dec 2008
Posts: 249
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Quote:
Originally Posted by Dragonballful23 View Post
When is the following true?

3^x+1=9^x
Did you mean 3^(x+1) = 9^x?

If so, then:

3^(x+1) = (3^2)^x
3^(x+1) = 3^(2x)
x+1 = 2x
x = 1

But the way the problem is originally written is much harder to solve.

(3^x) + 1 = 9^x
(3^x) + 1 = 3^(2x)
1 = 3^(2x) - 3^x
3^(2x) - 3^x - 1 = 0

Let a = 3^x

a^2 - a - 1 = 0
Then using the quadratic formula a = (1 + sqrt(5))/2 which is the golden ratio!

So 3^x = (1 + sqrt(5))/2
Taking log base 3 (log3) of both sides gives:
x = log3[(1 + sqrt(5))/2]

or using natural logs x = [ln(1 + sqrt(5)) - ln(2)]/(ln(3))

Last edited by MAS1; 03-12-2015 at 09:37 AM.. Reason: Solution to problem
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