SOLUTION: log16(9x+5)-log16(x^2-1)=1/2

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Question 401277: log16(9x+5)-log16(x^2-1)=1/2
Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
log%2816%2C+%289x%2B5%29%29-log%2816%2C+%28x%5E2-1%29%29=1%2F2
Solving a equation where the variable is in the argument (or base) of a logarithm usually starts with transforming the equation into one of the following forms:
log(expression) = other-expresison
or
log(expression) = log(other-expresison)

Your equation, with its "non-log" term of 1/2, will be easier to transform into the first form above.

To achieve the first form all we need to do is to combine the two logarithms into one. They are not like terms so we cannot subtract them. (Like terms involving logarithms have logarithms of the same base and the same argument Yours have the same base but different arguments.

Even though they cannot be subtracted, there is a property of logarithms, log%28a%2C+%28p%29%29+-+log%286%2C+%28q%29%29+=+log%28a%2C+%28p%2Fq%29%29, that will allow us to combine the two logarithms. This property requires:
  • Logarithms of the same base; and
  • A "-" between the two logarithms; and
  • Coefficients of 1 one the logarithms.
Your logarithms meet all three requirements so we can go ahead and use the property to combine the logarithms:
log%2816%2C+%28%289x%2B5%29%2F%28x%5E2-1%29%29%29=1%2F2
We now have the first form. With this form the next step is to rewrite the equation in exponential form. In general log%28a%2C+%28p%29%29+=+q is equivalent to p+=+a%5Eq. Using this pattern on your equation we get:
%289x%2B5%29%2F%28x%5E2-1%29=16%5E%281%2F2%29
which, simce 1/2 as an exponent means square root, simplifies to:
%289x%2B5%29%2F%28x%5E2-1%29=4
We now have an equation we can solve. Multiplying both sides by %28x%5E2-1%29 (to eliminate the fraction) we get:
%28%289x%2B5%29%2F%28x%5E2-1%29%29%28x%5E2-1%29=%284%29%28x%5E2-1%29
which simplifies to:
9x%2B5=4x%5E2-4
This is a quadratic equation so we want one side to be zero. Subtracing 9x and 5 from each side we get:
0=4x%5E2-9x-9
Next we factor (or use the Quadratic Formula). This factors fiarly easily:
(4x+3)(x-3) = 0
From the Zero Product Property we know that this (or any) product can be zero only if one (or more) of the factors is zero. So:
4x+3 = 0 or x-3 = 0
Solving these we get:
x = -3/4 or x = 3

When solving logarithmic equations like yours, you must check you answer(s)! You must ensure that all arguments (and bases) of all logarithms remain positive when the variable is equal to a "solution". If an argument (or base) works out to be zero or negative you must reject that "solution". A zero or negative argument (or base) can happen even if no mistakes have been made! This is why it is required to check.

Always use the original equation to check:
log%2816%2C+%289x%2B5%29%29-log%2816%2C+%28x%5E2-1%29%29=1%2F2
Checking x = -3/4:
log%2816%2C+%289%28-3%2F4%29%2B5%29%29-log%2816%2C+%28%28-3%2F4%29%5E2-1%29%29=1%2F2
which simplifies as follows:
log%2816%2C+%28-27%2F4%2B5%29%29-log%2816%2C+%289%2F16-1%29%29=1%2F2
log%2816%2C+%28-27%2F4%2B20%2F4%29%29-log%2816%2C+%289%2F16-16%2F16%29%29=1%2F2
log%2816%2C+%28-7%2F4%29%29-log%2816%2C+%28-7%2F16%29%29=1%2F2
We can see that both arguments are negative. So we must reject this solution. (If even only one argument had been negative (or zero) we would still reject this solution.) Important: Rejecting this solution does not mean we made a mistake earlier!

Checking x = 3:
log%2816%2C+%289%283%29%2B5%29%29-log%2816%2C+%28%283%29%5E2-1%29%29=1%2F2
which simplifies as follows:
log%2816%2C+%2827%2B5%29%29-log%2816%2C+%289-1%29%29=1%2F2
log%2816%2C+%2832%29%29-log%2816%2C+%288%29%29=1%2F2
We can see that both arguments (and bases) are positive. So there is no reason to reject this solution. We have completed the required part of the check. The rest of the check will tell us if we made a mistake. You are welcome to finish the check.

So the only solution to your equation is x = 3.