SOLUTION: How do I express the terms of a natural logarithm? The answer has to be a solution set. 7^(3x+2)=2^(x+3) Thanks!

Question 378219: How do I express the terms of a natural logarithm?
The answer has to be a solution set.
7^(3x+2)=2^(x+3)
Thanks!
Answer by jsmallt9(3758) (Show Source): You can put this solution on YOUR website!

Solving equations where the variable is in an exponent normally involves use of logarithms. Usually you can choose any base of logarithm. But you were instructed to use natural logarithms (ln). So we will find the natural logarithm of each side:

Now we use a property of logarithms, , to "move" the exponents of the arguments out in front. It is this very property that is the reason we use logarithms on equations like this. It allows us to "move" the exponent, where the variable is, to a location where we can use "regular" Algebra to solve the equation. Using this property on your equation we get:
(3x+2)*ln(7) = (x+3)*ln(2)
On each side we can use the Distributive Property to simplify:
3x*ln(7) + 2*ln(7) = x*ln(2)+3*ln(2)
Next we gather the x terms on one side and the other terms on the other side of the equation. Subtracting x*ln(2) and 2*ln(7) from each side we get:
3x*ln(7) - x*ln(2) = 3*ln(2) - 2*ln(7)
Now we can factor out x on the left side:
x*(3*ln(7) - ln(2)) = 3*ln(2) - 2*ln(7)
And finally we divide both sides of the equation by (3ln(7) - ln(2)):

This is an exact expression for the solution to your equation. The solution set is this single number. If you want a decimal approximation, then you need to get out your calculator, find the logarithms and simplify the expression.
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