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

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Question 609727: log(base 4)(4b+14)-log(base4)(b^2-3b-17)=1/2
Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
log%284%2C+%284b%2B14%29%29-log%284%2C+%28b%5E2-3b-17%29%29=1%2F2
You want "log(expression) = number". So we will start by using a property of logs, log%28a%2C+%28p%29%29+-+log%28a%2C+%28q%29%29+=+log%28a%2C+%28p%2Fq%29%29, to combine the two logs:
log%284%2C+%28%284b%2B14%29%2F%28b%5E2-3b-17%29%29%29=1%2F2

Now that we have the desired form. The next step is to rewrite the equation in exponential form. In general log%28a%2C+%28p%29%29+=+q is equivalent to a%5Eq+=+p. Using this pattern on our equation we get:
4%5E%281%2F2%29+=+%284b%2B14%29%2F%28b%5E2-3b-17%29
Since 1/2 as an exponent means square root and since the square root of 4 is 2, the left side is a 2:
2+=+%284b%2B14%29%2F%28b%5E2-3b-17%29

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:
%28b%5E2-3b-17%29%2A%282%29+=+%28b%5E2-3b-17%29%2A%28%284b%2B14%29%2F%28b%5E2-3b-17%29%29
which simplifies to:
2b%5E2-6b-34+=+4b%2B14
Since this is a quadratic equation we want one side to be zero. Subtracting 4b and 14 from each side:
2b%5E2-10b-48+=+0
Now we factor. First the GCF:
2%28b%5E2-5b-24%29+=+0
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:
log%284%2C+%284b%2B14%29%29-log%284%2C+%28b%5E2-3b-17%29%29=1%2F2
Checking b = 8:
log%284%2C+%284%288%29%2B14%29%29-log%284%2C+%28%288%29%5E2-3%288%29-17%29%29=1%2F2
Simplifying:
log%284%2C+%2832%2B14%29%29-log%284%2C+%2864-3%288%29-17%29%29=1%2F2
log%284%2C+%2846%29%29-log%284%2C+%2864-24-17%29%29=1%2F2
log%284%2C+%2846%29%29-log%284%2C+%2823%29%29=1%2F2
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:
log%284%2C+%284%28-3%29%2B14%29%29-log%284%2C+%28%28-3%29%5E2-3%28-3%29-17%29%29=1%2F2
Simplifying:
log%284%2C+%28-12%2B14%29%29-log%284%2C+%289-3%28-3%29-17%29%29=1%2F2
log%284%2C+%282%29%29-log%284%2C+%289%2B9-17%29%29=1%2F2
log%284%2C+%282%29%29-log%284%2C+%281%29%29=1%2F2
Again both arguments are positive. So b = -3 checks out, too.

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