Question 372991: log2(3x+2) - log4(x) =3
Answer by jsmallt9(3758)   (Show Source):
You can put this solution on YOUR website! 
For equations like this, with the variable in the argument of one of more logarithms, you usually start by 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 3 on the right side, the second form will be more difficult to achieve. SO we will aim for the first form. This will require that we somehow express the two logarithms as one.
Two logarithms with a minus between them, like in your equation, are often combined into one using a property of logarithms:  . But this property requires coefficients of 1 (which your logarithms have) and the same bases (which your logarithms do not have). Unless we can figure out how to get the bases the same, we will not be able to proceed.
Fortunately there is a base conversion formula:
which allows us to convert a base a logarithm into an expression of base b logarithms. With your logarithms we are going to use this formula to convert the base 4 logarithm into and expression of base 2 logarithms. (We could do it the other way around, too, but since 4 is an obvious (I hope) power of 2 this way will be simpler.) So
Since  ,  . So the expression above simplifies as follows:
Replacing the base 4 log in the equation with this expression we get:
We now have the bases the same! But the coefficients are no longer 1's!? Fortunately we have a property of logarithms,  , which allows us to move a coefficient of a logarithm into the argument as an exponent. Using this on your equation we get:
Since a power of 1/2 means square root, I am going to rewrite that argument as a square root:
We can finally use the first property (from above) to combine the two logarithms into one:
And we finally have the first form (log(expression) = other-expression). With this form we proceed by rewriting the equation in exponential form. In general  is equivalent to  . Using this on your equation we get:
which simplifies to:
The logarithms are gone. To solve what's left we will start by multiplying both sides by the square root to get rid of the fraction:
Next we square both sides to get rid of the square root:
(Be careful squaring the left side. Exponents do not distribute! Use FOIL or the pattern for (a+b)^2.)
With a quadratic equation we want one side zero so we'll subtract 64x from each side:
This will not factor so we will need to use the quadratic formula:
which simplifies as follows:
In long form this is:
 or 
You must check these answers. You original equation is a logarithmic equation and solutions for these must be checked to make sure no arguments (or bases) of any logarithms become zero or negative! Also, whenever you square both sides of an equation, like we did above, you must check for extraneous solutions. (Extraneous solutions are solutions which work in the squared equation but not in the original equation. Either of these reasons by themselves require a check. And we have both!
I will leave the checking to you. Use the original equation to check and you probably will want to use your calculator to calculate decimals to use for x.
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