SOLUTION: log(base 4)(4b+14)-log(base4)(b^2-3b-17)=1/2

Question 609727: log(base 4)(4b+14)-log(base4)(b^2-3b-17)=1/2
Answer by jsmallt9(3758) (Show Source): You can put this solution on YOUR website!

You want "log(expression) = number". So we will start by using a property of logs, , to combine the two logs:

Now that we have the desired form. The next step is to rewrite the equation in exponential form. In general is equivalent to . Using this pattern on our equation we get:

Since 1/2 as an exponent means square root and since the square root of 4 is 2, the left side is a 2:

Now that the variable is "out in the open", we can solve for it. First let's get rid of the fraction. Multiplying both sides by the denominator:

which simplifies to:

Since this is a quadratic equation we want one side to be zero. Subtracting 4b and 14 from each side:

Now we factor. First the GCF:

Now the trinomial:
2(b-8)(b+3) = 0
From the Zero Product Property we know that one (or more of these factors must be zero. Since the 2 is not zero:
b-8 = 0 or b+3 = 0
Solving these we get:
b = 8 or b = -3

Checking answers to logarithmic equations is not optional! You must at least ensure that the proposed solutions make the arguments positive. Any "solution" that makes an argument to a logarithm zero or negative must be rejected since arguments of logs can never be zero or negative. Use original equation to check:

Checking b = 8:

Simplifying:

We can now see that both arguments are positive. (The rest of the check is optional. You're welcome to finish the check.) So b = 8 checks out.

Checking b = -3:

Simplifying:

Again both arguments are positive. So b = -3 checks out, too.

So your equation has two solutions:
b = 8 or b = -3

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