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We want the equation in the form:
log(expression) = other-expression
So somehow we need to combine the two logarithms into one. These two logarithms are not like terms so we cannot subtract them. But there is a property of logarithms, , which can be used to combine two logarithms if al of the following are true:
there is a "-" between them
the bases of the logarithms are the same
the coefficients of the logarithms are 1's
Your logarithms meet the first two but not the last. So now our goal is to get rid of the 2 in front of the first log. And fortunately there is another property of logarithms, , which can be used to move a coefficient of a logarithm into its argument as an exponent. Using this on your first log we get:
These are still not like terms so we still cannot subtract them. But we can now use the other property to combine them:
We now have the desired form. Once we have this form the next step is to rewrite the equation in exponential form:
which simplifies to:
Now the variable is out of the argument where we can "get at it". Solving this for x we start by multiplying both sides by 5 to get rid of the fraction:
So or
Simplifying these square roots we get: or
When solving logarithmic equations, it is important (not just a good idea) to check your answers. Even if we've done everything correct so far, we need to make sure that each answer makes the argument of any logarithms positive.
Always use the original equation to check:
Checking
Both arguments are positive so it looks good. (You're welcome to finish the check on your own.)
Checking
As you can see the argument to the first logarithm is negative. We cannot allow this to occur. So we reject this solution.
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