Question 345890
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)<br>
Now we will replace f(x) with {{{3^x}}}:
{{{1/2 = 3^x}}}
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!<br>
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(3, (1/2)) = log(3, (3^x))}}}
On the right side we can use a property of logarithms, {{{log(a, (p^q)) = q*log(a, (p))}}}, 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(3, (1/2)) = x*log(3, (3))}}}
By definition, {{{log(3, (3)) = 1}}} so now we have:
{{{log(3, (1/2)) = x*1}}}
{{{log(3, (1/2)) = x}}}
And we have an exact expression for the answer.<br>
If we need a decimal approximation for the answer, we could:
1) Use the base conversion formula, {{{log(a, (p)) = log(b, (p))/log(b, (a))}}} on the answer we got above to convert it to a base our calculators "know" (base 10 or base e (ln)):
{{{x = log(3, (1/2)) = ln(1/2)/ln(3)}}}
or
{{{x = log(3, (1/2)) = log((1/2))/log((3))}}}
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!<br>
or<br>
2) Use base 10 or base e logarithms from the start:
{{{1/2 = 3^x}}}
{{{ln(1/2) = ln(3^x)}}}
{{{ln(1/2) = x*ln(3)}}}
Divide both sides by ln(3):
{{{ln(1/2)/ln(3) = 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.