SOLUTION: Find the exact solution, using common logarithms. 4^x+64(4^-x)=20

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Question 354654: Find the exact solution, using common logarithms.
4^x+64(4^-x)=20
Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
4%5Ex%2B64%2A4%5E%28-x%29=20
Rewriting this with positive exponents we get:
4%5Ex%2B64%2F4%5Ex=20
Now we can eliminate the fraction by multiplying both sides by 4%5Ex:
4%5Ex%2A%284%5Ex%2B64%2F4%5Ex%29=+4%5Ex%2A%2820%29
Using the Distributive Property on the left sides we get:
4%5Ex%2A4%5Ex%2B4%5Ex%2A%2864%2F4%5Ex%29=+4%5Ex%2A%2820%29
For 4%5Ex%2A4%5Ex the rule is to add the exponents. so 4%5Ex%2A4%5Ex+=+4%5E%28x%2Bx%29+=+4%5E%282x%29. In the second term the 4%5Ex's cancel. So the equation is now:
4%5E%282x%29+%2B+64+=+20%2A4%5Ex
Subtracting 20%2A4%5Ex from each side we have:
4%5E%282x%29+-+20%2A4%5Ex+%2B+64+=+0
Since the exponent on 4%5E%282x%29 is twice the exponent of 4%5Ex, then this equation is in quadratic form for 4%5Ex. So we can solve it like a quadratic equation. We can factor the left side:
%284%5Ex+-16%29%284%5Ex+-+4%29+=+0
From the Zero Product Property we know that this (or any) product can be zero only if on of the factors is zero. So:
4%5Ex+-16+=+0 or 4%5Ex+-+4+=+0
Adding 16 to both sides of the first equation and 4 to both sides of the second equation we get:
4%5Ex+=+16 or 4%5Ex+=+4
At this point we could mentally figure out the solutions. (What power of 4 is 16? and what power of 4 is 4? These are very simple questions to answer.) But since you were told to use common logarithms we will do so:
log%28%284%5Ex%29%29+=+log%28%2816%29%29 or log%28%284%5Ex%29%29+=+log%28%284%29%29
Now we can use the property of logarithms, log%28a%2C+%28p%5Eq%29%29+=+q%2Alog%28a%2C+%28p%29%29 to move the exponents in the arguments out front:
x%2Alog%28%284%29%29+=+log%28%2816%29%29 or x%2Alog%28%284%29%29+=+log%28%284%29%29
Then we can divide both sides of the both equations by log(4):
x+=+log%28%2816%29%29%2Flog%28%284%29%29 or x+=+1
If you use your calculator on the logs in first equation and then divide them you should get 2 (or something extremely close to 2).