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

Question 354654: Find the exact solution, using common logarithms.
4^x+64(4^-x)=20
Answer by jsmallt9(3758) (Show Source): You can put this solution on YOUR website!

Rewriting this with positive exponents we get:

Now we can eliminate the fraction by multiplying both sides by :

Using the Distributive Property on the left sides we get:

For the rule is to add the exponents. so . In the second term the 's cancel. So the equation is now:

Subtracting from each side we have:

Since the exponent on is twice the exponent of , then this equation is in quadratic form for . So we can solve it like a quadratic equation. We can factor the left side:

From the Zero Product Property we know that this (or any) product can be zero only if on of the factors is zero. So:
or
Adding 16 to both sides of the first equation and 4 to both sides of the second equation we get:
or
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:
or
Now we can use the property of logarithms, to move the exponents in the arguments out front:
or
Then we can divide both sides of the both equations by log(4):
or
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).
RELATED QUESTIONS
Find the exact solution using common logarithms. log(x^2 + 4) − log(x + 2) = 2... (answered by ewatrrr)

Find the exact solution, using common logarithms:... (answered by josgarithmetic)

Find the exact solution, using common logarithms.... (answered by jsmallt9)

Find the exact solution, using common logarithms. log(x − 3) − log(5x − 7) =... (answered by josgarithmetic,greenestamps,ikleyn)

I need help with this problem: Find the solution using common logarithms, and... (answered by josgarithmetic)

Find the exact solution, using common logarithms, and a two-decimal-place approximation,... (answered by lwsshak3)

Find the exact solution, using common logarithms. (Enter your answers as a... (answered by Theo,MathTherapy)

Find the approximate solution to the nearest hundreth using common logarithms... (answered by stanbon)

Find the exact solution, using common logarithms. (Enter your answers as a... (answered by Alan3354)