Question 609727
{{{log(4, (4b+14))-log(4, (b^2-3b-17))=1/2}}}
You want "log(expression) = number". So we will start by using a property of logs, {{{log(a, (p)) - log(a, (q)) = log(a, (p/q))}}}, to combine the two logs:
{{{log(4, ((4b+14)/(b^2-3b-17)))=1/2}}}<br>
Now that we have the desired form. The next step is to rewrite the equation in exponential form. In general {{{log(a, (p)) = q}}} is equivalent to {{{a^q = p}}}. Using this pattern on our equation we get:
{{{4^(1/2) = (4b+14)/(b^2-3b-17)}}}
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 = (4b+14)/(b^2-3b-17)}}}<br>
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:
{{{(b^2-3b-17)*(2) = (b^2-3b-17)*((4b+14)/(b^2-3b-17))}}}
which simplifies to:
{{{2b^2-6b-34 = 4b+14}}}
Since this is a quadratic equation we want one side to be zero. Subtracting 4b and 14 from each side:
{{{2b^2-10b-48 = 0}}}
Now we factor. First the GCF:
{{{2(b^2-5b-24) = 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<br>
Checking answers to logarithmic equations is <i>not optional!</i> 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 <i>never</i> be zero or negative. Use original equation to check:
{{{log(4, (4b+14))-log(4, (b^2-3b-17))=1/2}}}
Checking b = 8:
{{{log(4, (4(8)+14))-log(4, ((8)^2-3(8)-17))=1/2}}}
Simplifying:
{{{log(4, (32+14))-log(4, (64-3(8)-17))=1/2}}}
{{{log(4, (46))-log(4, (64-24-17))=1/2}}}
{{{log(4, (46))-log(4, (23))=1/2}}}
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.<br>
Checking b = -3:
{{{log(4, (4(-3)+14))-log(4, ((-3)^2-3(-3)-17))=1/2}}}
Simplifying:
{{{log(4, (-12+14))-log(4, (9-3(-3)-17))=1/2}}}
{{{log(4, (2))-log(4, (9+9-17))=1/2}}}
{{{log(4, (2))-log(4, (1))=1/2}}}
Again both arguments are positive. So b = -3 checks out, too.<br>
So your equation has two solutions:
b = 8 or b = -3<br>