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Question 47776: Find the exact solution to the equation 3^x+5 = 9^x. I am having difficulty working this one out. Please be so kind as to help me with this problem.
Found 2 solutions by pizza, stanbon:
Answer by pizza(14)   (Show Source):
You can put this solution on YOUR website! Do you mean  or  ?
Let me first assume it is the second statement that you have problem solving. In algebra, you play with equations, and every equation has a left hand side and right hand side. You can do almost anything to the left hand side as long as you do the exact same thing to the right hand side.
In the question, we start with 3x + 5 = 9x.
First, we subtract 3x from both sides, so as to minimise the appearance of x.
That is, 3x+5 -3x = 9x - 3x
which is 5 = 6x
Next, to remove the 6, divide both sides by 6
That is 5/6 = 6x/6
which is 5/6 = x
You can then read off the answer, x = 5/6.
Answer by stanbon(75887) (Show Source):
You can put this solution on YOUR website! 3^x+5 = 9^x
9^x-3^x-5=0
(3^x)^(2) - 3^x -5 = 0
This is a quadratic where the variable is 3^x
Let 3^x = w, then rewrite the problem as:
w^2 - w -5 =0
Use the quadratic formula to get the following:
w=[1+-sqrt(1+20)]/2
w=[1+-sqrt(21)]/2
Now, reverting back to the fact that w=3^x you get:
3^x=[1+-sqrt(21)]/2
3^x=2.79128785... or 3^x is a negative number( which it cannot be)
Take the log to get:
xlog3 =2.79128785...
The x=5.85026934...
Cheers,
Stan H.
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