document.write( "Question 609727: log(base 4)(4b+14)-log(base4)(b^2-3b-17)=1/2 \n" ); document.write( "
Algebra.Com's Answer #383967 by jsmallt9(3758)\"\" \"About 
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\"
\n" ); document.write( "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:
\n" ); document.write( "\"log%284%2C+%28%284b%2B14%29%2F%28b%5E2-3b-17%29%29%29=1%2F2\"

\n" ); document.write( "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:
\n" ); document.write( "\"4%5E%281%2F2%29+=+%284b%2B14%29%2F%28b%5E2-3b-17%29\"
\n" ); document.write( "Since 1/2 as an exponent means square root and since the square root of 4 is 2, the left side is a 2:
\n" ); document.write( "\"2+=+%284b%2B14%29%2F%28b%5E2-3b-17%29\"

\n" ); document.write( "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:
\n" ); document.write( "\"%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\"
\n" ); document.write( "which simplifies to:
\n" ); document.write( "\"2b%5E2-6b-34+=+4b%2B14\"
\n" ); document.write( "Since this is a quadratic equation we want one side to be zero. Subtracting 4b and 14 from each side:
\n" ); document.write( "\"2b%5E2-10b-48+=+0\"
\n" ); document.write( "Now we factor. First the GCF:
\n" ); document.write( "\"2%28b%5E2-5b-24%29+=+0\"
\n" ); document.write( "Now the trinomial:
\n" ); document.write( "2(b-8)(b+3) = 0
\n" ); document.write( "From the Zero Product Property we know that one (or more of these factors must be zero. Since the 2 is not zero:
\n" ); document.write( "b-8 = 0 or b+3 = 0
\n" ); document.write( "Solving these we get:
\n" ); document.write( "b = 8 or b = -3

\n" ); document.write( "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:
\n" ); document.write( "\"log%284%2C+%284b%2B14%29%29-log%284%2C+%28b%5E2-3b-17%29%29=1%2F2\"
\n" ); document.write( "Checking b = 8:
\n" ); document.write( "\"log%284%2C+%284%288%29%2B14%29%29-log%284%2C+%28%288%29%5E2-3%288%29-17%29%29=1%2F2\"
\n" ); document.write( "Simplifying:
\n" ); document.write( "\"log%284%2C+%2832%2B14%29%29-log%284%2C+%2864-3%288%29-17%29%29=1%2F2\"
\n" ); document.write( "\"log%284%2C+%2846%29%29-log%284%2C+%2864-24-17%29%29=1%2F2\"
\n" ); document.write( "\"log%284%2C+%2846%29%29-log%284%2C+%2823%29%29=1%2F2\"
\n" ); document.write( "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.

\n" ); document.write( "Checking b = -3:
\n" ); document.write( "\"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\"
\n" ); document.write( "Simplifying:
\n" ); document.write( "\"log%284%2C+%28-12%2B14%29%29-log%284%2C+%289-3%28-3%29-17%29%29=1%2F2\"
\n" ); document.write( "\"log%284%2C+%282%29%29-log%284%2C+%289%2B9-17%29%29=1%2F2\"
\n" ); document.write( "\"log%284%2C+%282%29%29-log%284%2C+%281%29%29=1%2F2\"
\n" ); document.write( "Again both arguments are positive. So b = -3 checks out, too.

\n" ); document.write( "So your equation has two solutions:
\n" ); document.write( "b = 8 or b = -3
\n" ); document.write( "
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