document.write( "Question 613652: Solve each equation. Show all your work. Round your answers to four decimal places.
\n" ); document.write( " a. 9^4x=81
\n" ); document.write( " b. log4(9x+5)-log4(2x+2)=1 \n" ); document.write( "

Algebra.Com's Answer #386139 by jsmallt9(3758) $\"\"$ $\"About$

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a) $\"9%5E%284x%29+=+81\"$
\n" ); document.write( "Often one uses logarithms to solve equations where the variable is in an exponent. The exception is when you can rewrite the equation so that it says that two powers of the same number are equal. Since $\"81+=+9%5E2\"$ this equation can easily be rewritten as one power of 9 equals another power of 9:
\n" ); document.write( " $\"9%5E%284x%29+=+9%5E2\"$

\n" ); document.write( "The next step in this situation is simple and it is based on the idea that the only way two powers of 9 can be equal is if the exponents are equal. So
\n" ); document.write( "4x = 2
\n" ); document.write( "Solving this is very easy. Divide by 4:
\n" ); document.write( "x = 1/2

\n" ); document.write( "b) $\"log%284%2C%289x%2B5%29%29-log%284%2C%282x%2B2%29%29=1\"$
\n" ); document.write( "Solving equations where the variable is in the argument of a logarithm usually starts with using algebra and/or properties of logarithms to transform the equation into one of the following general forms:
\n" ); document.write( "log(expression) = other-expression
\n" ); document.write( "or
\n" ); document.write( "log(expression) = log(other-expression)

\n" ); document.write( "With your equation's \"non-log\" term of 1, it will be more difficult to transform it into the second, \"all-log\" form. So we will aim for the first form.

\n" ); document.write( "If we can find a way to combine the two logarithms into one then we will have the first form. The two logs are not like terms so we cannot simply subtract them into one term. (Llike logarithmic terms have the same bases and the same arguments. Your logs have the same bases but the arguments are different.)

\n" ); document.write( "Fortunately there is another way we can combine logarithmic terms. Two properties of logarithms can be used to combine two logarithms into one:

$\"log%28a%2C+%28p%29%29+%2B+log%28a%2C+%28q%29%29+=+log%28a%2C+%28p%2Aq%29%29\"$
$\"log%28a%2C+%28p%29%29+-+log%28a%2C+%28q%29%29+=+log%28a%2C+%28p%2Fq%29%29\"$

These properties require that the logs have the same base and that the coefficients of the logs be 1's. The first property is used to combine terms that have a \"+\" between them while the second one is used to combine terms when there is a \"-\" between the terms.

\n" ); document.write( "Your logs meet requirements of these properties and since there is a \"-\" between them we will use the second property:
\n" ); document.write( " $\"log%284%2C%28%289x%2B5%29%2F%282x%2B2%29%29%29=1\"$
\n" ); document.write( "We now have the first form.

\n" ); document.write( "The next step with the first form 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( " $\"4%5E1+=+%289x%2B5%29%2F%282x%2B2%29\"$
\n" ); document.write( "which simplifies to:
\n" ); document.write( " $\"4+=+%289x%2B5%29%2F%282x%2B2%29\"$

\n" ); document.write( "Now that the logarithms are gone we can use \"regular\" algebra to solve for x. Multiplying both sides by 2x+2 we get:
\n" ); document.write( "8x + 8 = 9x + 5
\n" ); document.write( "Subtracting 8x from each side:
\n" ); document.write( "8 = x + 5
\n" ); document.write( "Subtracting 5 from each side:
\n" ); document.write( "3 = x

\n" ); document.write( "When solving equations like $\"log%284%2C%289x%2B5%29%29-log%284%2C%282x%2B2%29%29=1\"$ , you must check your answer(s). It is not optional! You must at least check to make sure that the answer(s) make all arguments positive. Any \"answer\" that makes an argument zero or negative must be rejected. Use the original equation to check:
\n" ); document.write( " $\"log%284%2C%289x%2B5%29%29-log%284%2C%282x%2B2%29%29=1\"$
\n" ); document.write( "Checking x = 3:
\n" ); document.write( " $\"log%284%2C%289%283%29%2B5%29%29-log%284%2C%282%283%29%2B2%29%29=1\"$
\n" ); document.write( "You can probably see already that both arguments are going to work out to be positive numbers. (If you can't see this, then continue to simplify the arguments until you can see that they are positive.) We have completed the required portion of the check and this solution passes. So your answer is x = 3. \n" ); document.write( "

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