SOLUTION: Use Logarithms to find an exact expression for x if 5^-3x + 3 = 7

Question 353514: Use Logarithms to find an exact expression for x if
5^-3x + 3 = 7
Answer by jsmallt9(3758) (Show Source): You can put this solution on YOUR website!

First solve for the term with the variable:

Now we use logarithms (for reasons which will become clear shortly). The base we use for the logarithms does not matter in any important way. But as we will see the answer will be simpler if we use base 5:

Now we can use a property of logarithms, , to move the exponent out in front.

This property of logarithms, with its ability to extract an exponent, is the very reason we use logarithms! The variable is now "where we can get at it".
By definition, log(5, (5)) = 1. (This is why using base 5 is going to make our answer simpler.) This gives us:

Now we can divide by -3 (or multiply both sides by -1/3:

This is an exact expression for x.

If we had used a different base logarithm, like base 10, on

we would get:

This time the first log doesn't simplify. To solve for x we can divide both sides by :

This is another, less simple but still exact expression for x. Although less simple, this version has the advantage of using logarithms your calculator "knows". So if you want a decimal approximation, this version will be easier to use.
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