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With the variable in the argument of a logarithm like this, you often start the solution by transforming the equation into one of the following forms:
log(expression) = other-expression
or
log(expression) = log(other-expression)
With the 2 on the right side, it will be a little difficult to get an "all logarithm" equation like the second form. So we will aim for the first one. This means we will need to combine our two logarithms into one. We cannot just subtract the two logarithms we have because they are not like terms. Fortunately we have some properties of logarithms that will allow us to combine these two:
Both of these properties require a coefficient of 1 in front of the logarithm but we have yet another property, , which gives us a way to eliminate coefficients that are not 1.
We'll start by using this last property to eliminate the 2 in front of the first logarithm:
Now our coefficients are 1's and we can combine the two logarithms. We'll use the second property it handles a subtraction of logarithms and that is what we have:
Now we have the first form. The next step with this form is to rewrite the equation in exponential form. To do this we have to remember that is equivalent to . Using this on our equation we get:
which simplifies to:
Now we have a quadratic equation to solve. We'll get rif of the fraction first by multiplying both sides by 3:
Next we'll find the square root of each side:
Normally we would use for so we could account for negative x's. But in this equation, x has to be positive. x has to be positive because it is an argument to a logarithm in the original equation and arguments to all logarithms must be positive. Since x must be positive we will just use x for :
Now we just simplify the square root:
This is the simplified, exact answer.
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