SOLUTION: log(3x)+log(x)=8 its in logarithm form i know how to get the answer for x i just dont how to get this equation from a log to exponential form

Question 393614: log(3x)+log(x)=8 its in logarithm form i know how to get the answer for x i just dont how to get this equation from a log to exponential form
Answer by jsmallt9(3758) (Show Source): You can put this solution on YOUR website!

I'd be curious to know how you can solve for x without knowing how to write this equation in exponential form.

Solving logarithmic equations 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)

With the "non-log" term of 8 it will be more difficult to reach the second form. So we will aim for the first form. We want one side of the equation to be a single logarithm. For this we can use the property of logarithms: . Using this on your equation we get:

which simplifies to:

We now have the first form. The next step with the first form is to rewrite the equation in exponential form. In general } is equivalent to . Using this pattern on your equation we get:
(since the base of "log" is 10)
which simplifies to:

This equation we can solve. Dividing both sides by 3 we get:

Now we find the square root of each side:

(Note: Algebra.com's software will not let me use the "plus or minus" symbol without something in front of it. This is why the extra zero is there in front.)
In long form this is:
or

With any logarithmic equation like yours you must check your answers. You must ensure that no arguments (or bases) of any logarithms become negative or zero. This can happen even if no mistakes were made! So even if you make no errors, you still have to check for this. And if you find that an "answer" does make an argument (or base) negative or zero then you must reject that "answer".

Always use the original equation to check:

Checking :

We can see already that both arguments will be positive. So there is no reason to reject this solution. We have done the required part of the check for this solution. The rest of the check will just tell us if we made an error. You are welcome to finish the check if you like.

Checking :

We can already see that both arguments are going to be negative, So we must reject this solution. (If even just one argument had been negative or zero we would still reject the solution.

So the only solution to you equation is: . This square root is not in proper form. (There should not be a fraction in a square root. So we should rationalize the denominator:

And then simplify the numerator:

This is the proper form for your solution.
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