Question 610651
log(x-3) + log(x+3) = 4
Solving equation usually start with using algebra and/or log properties to transform the equation into one of the following forms:
log(expression) = other-expression
or
log(expression) = log(other-expression)<br>
With the "non-log" term of 4 the "all-log" second form will be more difficult to achieve. So we will aim for the first form. If we could combine the two logarithms into one then we would have the first form. And there happens to be a property of logarithms, {{{log(a, (p)) + log(a, (q)) = log(a, (p*q))}}}, that will allow us to do just that:
log((x-3)*(x+3)) = 4
which simplifies to:
{{{log((x^2-9)) = 4}}}<br>
With the first form the next step is to rewrite the equation in exponential form. In general {{{log(a,(p)) = q}}} is equivalent to {{{a^q = p}}}. Using this pattern on our equation we get:
{{{10^4 = x^2 - 9}}}
which simplifies to:
{{{10000 = x^2 - 9}}}<br>
Now we solve for x. With a squared term but no first power terms we can just add 9:
{{{10009 = x^2}}}
... and find the square root of each side:
{{{x = sqrt(10009)}}} or {{{x = -sqrt(10009)}}}<br>
Last of all we check our answers. This is <i>not optional!</i> You must at least ensure that all arguments of all logarithms remain positive. Any "solution" that makes an argument of any logarithm zero or negative must be rejected. (These rejected "solutions", if any, are not a sign that a mistake was made.)<br>
Use the original equation to check:
{{{log((x-3)) + log((x+3)) = 4}}}
Checking {{{x = sqrt(10009)}}}:
{{{log(((sqrt(10009))-3)) + log(((sqrt(10009))+3)) = 4}}}
Since {{{sqrt(10009) > 100}}}, we can see that both arguments will be positive. This is the required part of the check. (You are welcome to finish the check to see if any mistakes were made.)<br>
Checking {{{x = -sqrt(10009)}}}:
{{{log(((-sqrt(10009))-3)) + log(((-sqrt(10009))+3)) = 4}}}
We can see that both arguments will be negative. So we must reject this "solution". (If only one argument had been negative (or zero) we would still reject the solution.)<br>
So the only solution to your equation is: {{{x = sqrt(10009)}}}