SOLUTION: (4)/(2+3^x)=1

Question 633316: (4)/(2+3^x)=1
Answer by jsmallt9(3758) (Show Source): You can put this solution on YOUR website!

First let's eliminate the fraction by multiplying both sides by {{2+3^x}}}:

Now we'll isolate the base and its exponent by subtracting 2 from each side:

The next step is to use logarithms. Any base of logarithm may be used. But there are advantages to choosing certain bases:

Choosing a base for the logarithm that matches th base of the exponent will result in the simplest possible expression for the solution. In this equation we would choose base 3 logarithms.
Choosing a base of logarithm you calculator "knows" (base 10 (log) or base e (ln)) will result in a less simple expression of the solution. But it will be an expression that can easily be converted to a decimal approximation.

I will do it both ways so you can see them both.

Using base 3 logarithms:

Next we use a property of logarithms, , which allows us to move the exponent of the argument out in front. It is this property that is the very reason we use usually logarithms to solve equations where the variable is in the exponent. The property allows us to move the exponent, where the variable is, out in front where "we can get at it" using "regular" algebra. Using this property we get:

Since this simplifies to just:

This is the simplest possible exact expression of the answer to your equation.

Using base e logarithms:

Using the property:

Unlike the base 3 log of 3, ln(3) does not just "disappear". To solve for x we must divide both sides by ln(3):

Not a simple as our earlier solution but this is another exact expression for the solution. And this one can easily be converted to a decimal approximation if needed/wanted.
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