Question 277826
log(x-9) = 1 - log(x)
Solving an 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-expression
or
log(expression) = log(other-expression)<br>
Since your equation has that "non-logarithmic" term (the 1), reaching the second form (which is all logarithms) will not be easy. So we will aim for the first form. We need the logarithms on one side so we'll start by adding log(x) to each side:
log(x-9) + log(x) = 1
Now we need to combine the logarithms into one. These are not like terms so we cannot add them. But there is a property of logarithms, {{{log(a, (p)) + log(a, (q)) = log(a, (p*q))}}}, which allows us to combine two logarithms with a "+" between them (as long as the bases are the same and the coefficients are 1's). Your bases are the same, 10, and the coefficients are 1's so we can use the property:
log((x-9)*(x)) = 1
which simplifies to:
{{{log((x^2 - 9x)) = 1}}}
We now have the equation in the first form. With this form we proceed by rewriting the equation in exponential form:
{{{x^2 - 9x = 10^1}}}
which simplifies to:
{{{x^2 - 9x = 10}}}
The variable is now out of the logarithms. This is a quadratic equation so we want one side equal to zero. So we'll subtract 10 from each side:
{{{x^2 - 9x - 10 = 0}}}
Now we factor it (or use the Quadratic Formula). This factors easily:
(x-10)(x-1) = 0
From the Zero Product Property we know that this product is zero only if one fo the factors is zero. So:
x-10 = 0 or x-1 = 0
Solving each of these we get:
x = 10 or x = 1<br>
When solving logarithmic equations, it is important (not just a good idea) to check your answers. Always use the original equation to check:
log(x-9) = 1 - log(x)
Checking x = 10:
log(10-9) = 1 - log(10)
log(1) = 1 - log(10)
Since log(1) = 0 and log(10) = 1 this simplifies to:
0 = 1 - 1
0 = 0 Check!<br>
Checking x = 1:
log(1-9) = 1 - log(1)
log(-8) = 1 - log(1)
At this point we have a problem. The first logarithm has a negative argument. Arguments of logarithms can never be negative or zero. So we must reject this solution.<br>
That makes x = 10 the only solution to your equation.