document.write( "Question 250052: Log(base 4) of X = Log (base 8) of 4x\r
\n" ); document.write( "\n" ); document.write( "Solve for x? \n" ); document.write( "
Algebra.Com's Answer #182979 by jsmallt9(3758)\"\" \"About 
You can put this solution on YOUR website!
\"log%284%2C+%28x%29%29+=+log%288%2C+%284x%29%29\"
\n" ); document.write( "The key to the solution is to recognize that 4 and 8 are both powers of 2. So we can rewrite each of these logarithms in terms of base 2 logarithms. You might be able to realize that since \"4+=+2%5E2\" and \"8+=+2%5E3\" and since logarithms are exponents, that \"%281%2F2%29log%282%2C+%28q%29%29+=+log%284%2C+%28q%29%29\" and \"%281%2F3%29log%282%2C+%28q%29%29+=+log%288%2C+%28q%29%29\". If this is hard to understand then here is some Algebra to show it. We will use a temporary variable:
\n" ); document.write( "Let \"z+=+log%284%2C+%28x%29%29\"
\n" ); document.write( "Rewriting this in exponential form we get:
\n" ); document.write( "\"4%5Ez+=+x\"
\n" ); document.write( "Replace 4 with \"2%5E2\":
\n" ); document.write( "\"%282%5E2%29%5Ez+=+x\"
\n" ); document.write( "Use the property of exponents, \"%28a%5Ep%29%5Eq+=+a%5E%28p%2Aq%29\":
\n" ); document.write( "\"2%5E%282z%29+=+x\"
\n" ); document.write( "Find the base 2 logarithm of each side:
\n" ); document.write( "\"log%282%2C+%282%5E%282z%29%29%29+=+log%282%2C+%28x%29%29\"
\n" ); document.write( "Using a property of logarithms, \"log%28a%2C+%28p%5Eq%29%29+=+q%2Alog%28a%2C+%28p%29%29\", we can move the exponent out front:
\n" ); document.write( "\"2z%2Alog%282%2C+%282%29%29+=+log%282%2C+%28x%29%29\"
\n" ); document.write( "Since \"log%282%2C+%282%29%29+=+1\" by definition:
\n" ); document.write( "\"2z+=+log%282%2C+%28x%29%29\"
\n" ); document.write( "Multiply both sides by 1/2:
\n" ); document.write( "\"z+=+%281%2F2%29log%282%2C+%28x%29%29\"
\n" ); document.write( "Replace our temporary variable with what it represents:
\n" ); document.write( "\"log%284%2C+%28x%29%29+=+%281%2F2%29log%282%2C+%28x%29%29\"
\n" ); document.write( "Similar logic shows that
\n" ); document.write( "\"log%288%2C+%284x%29%29+=+%281%2F3%29log%282%2C+%284x%29%29\"

\n" ); document.write( "So we can write our equation using base 2 logarithms:
\n" ); document.write( "\"%281%2F2%29log%282%2C+%28x%29%29+=+%281%2F3%29log%282%2C+%284x%29%29\"
\n" ); document.write( "Now we can use the property of logarithms used earlier, in the other direction, to move the coefficients back into the arguments as exponents:
\n" ); document.write( "\"log%282%2C+%28x%5E%281%2F2%29%29%29+=+log%282%2C+%28%284x%29%5E%281%2F3%29%29%29\"
\n" ); document.write( "Now that we have two base 2 logarithms that are equal, their arguments must be equal:
\n" ); document.write( "\"x%5E%281%2F2%29+=+%284x%29%5E%281%2F3%29\"
\n" ); document.write( "To solve this we'll raise both sides to the 6th power. (You'll see why in a minute.)
\n" ); document.write( "\"%28x%5E%281%2F2%29%29%5E6+=+%28%284x%29%5E%281%2F3%29%29%5E6\"
\n" ); document.write( "which simplifies to
\n" ); document.write( "\"x%5E3+=+%284x%29%5E2\"
\n" ); document.write( "(See why we used 6 now?)
\n" ); document.write( "Now we simplify
\n" ); document.write( "\"x%5E3+=+16x%5E2\"
\n" ); document.write( "and solve. With a cubed term, the way to solve this is to get one side equal to zero and factor. Subtract \"16x%5E2\" from each side:
\n" ); document.write( "\"x%5E3+-+16x%5E2+=+0\"
\n" ); document.write( "Factor out the GCF (which is \"x%5E2\"):
\n" ); document.write( "\"x%5E2%28x+-16%29+=+0\"
\n" ); document.write( "Now we can use the Zero Product Property which says that this product can be zero only if one of the factors is zero. So:
\n" ); document.write( "\"x%5E2+=+0\" or \"x-16+=+0\"
\n" ); document.write( "Solving each of these we get
\n" ); document.write( "\"x+=+0\" or \"x+=+16\"

\n" ); document.write( "With logarithmic equations we should always check our answers. We must make sure that the solutions do not make an argument to any logarithm zero or negative.
\n" ); document.write( "\"log%284%2C+%28x%29%29+=+log%288%2C+%284x%29%29\"
\n" ); document.write( "Checking x = 0:
\n" ); document.write( "\"log%284%2C+%280%29%29+=+log%288%2C+%284%280%29%29%29\"
\n" ); document.write( "As we can see, when x = 0 we get arguments that are zero so we have to reject x = 0 as a solution.
\n" ); document.write( "Checking x = 16:
\n" ); document.write( "\"log%284%2C+%2816%29%29+=+log%288%2C+%284%2816%29%29%29\"
\n" ); document.write( "which simplifies to
\n" ); document.write( "\"log%284%2C+%2816%29%29+=+log%288%2C+%2864%29%29\"
\n" ); document.write( "which simplifies to
\n" ); document.write( "\"2+=+2\" Check!

\n" ); document.write( "So the only solution is x = 16.\r
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