SOLUTION: how to solve, f(x)+1=3f(x) if f(x) is equal to 3^x

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Question 345890: how to solve, f(x)+1=3f(x) if f(x) is equal to 3^x
Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
f(x)+1 = 3f(x)
First we'll solve this for f(x). We'll start by subtracting f(x) from each side:
1 = 2f(x)
and then dividing both sides by 2:
1/2 = f(x)

Now we will replace f(x) with 3%5Ex:
1%2F2+=+3%5Ex
Next we'll solve for x. With an equation where the variable is in an exponent, you will usually use logarithms to solve. The question now is: What base of logarithm should be used? Answer: It doesn't really matter!

However in this problem, the simplest answer will be found if we use base 3 logarithms (because 3 is the base which has the exponent with x in it). So we will use base 3. Finding the base 3 logarithm of each side we get:
log%283%2C+%281%2F2%29%29+=+log%283%2C+%283%5Ex%29%29
On the right side we can use a property of logarithms, log%28a%2C+%28p%5Eq%29%29+=+q%2Alog%28a%2C+%28p%29%29, to move the exponent out in front. Using this property in this way is exactly the reason we use logarithms: We get to move the variable out of the exponent.
log%283%2C+%281%2F2%29%29+=+x%2Alog%283%2C+%283%29%29
By definition, log%283%2C+%283%29%29+=+1 so now we have:
log%283%2C+%281%2F2%29%29+=+x%2A1
log%283%2C+%281%2F2%29%29+=+x
And we have an exact expression for the answer.

If we need a decimal approximation for the answer, we could:
1) Use the base conversion formula, log%28a%2C+%28p%29%29+=+log%28b%2C+%28p%29%29%2Flog%28b%2C+%28a%29%29 on the answer we got above to convert it to a base our calculators "know" (base 10 or base e (ln)):
x+=+log%283%2C+%281%2F2%29%29+=+ln%281%2F2%29%2Fln%283%29
or
x+=+log%283%2C+%281%2F2%29%29+=+log%28%281%2F2%29%29%2Flog%28%283%29%29
I'll leave it up to you to use your calculator on these. Just be sure to find the two logarithms first, then divide. NOT divide first!

or

2) Use base 10 or base e logarithms from the start:
1%2F2+=+3%5Ex
ln%281%2F2%29+=+ln%283%5Ex%29
ln%281%2F2%29+=+x%2Aln%283%29
Divide both sides by ln(3):
ln%281%2F2%29%2Fln%283%29+=+x
This is an exact expression for the answer. Note: a) This is not as simple as we got using base 3 logarithms. b) The is the same expression we got above using the base conversion formula on our base 3 solution.