document.write( "Question 253695: If log(base14)7=a and log(base14)5=b, express log(base35)28 in terms of a and b. \n" ); document.write( "
Algebra.Com's Answer #188836 by jsmallt9(3758)\"\" \"About 
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\"log%2814%2C+%287%29%29+=+a\" and \"log%2814%2C+%285%29%29+=+b\"
\n" ); document.write( "Express \"log%2835%2C+%2828%29%29\" in terms of a and b.

\n" ); document.write( "This problem requires that we use regular Algebra and the properties of logarithms. Since we do not need any base 14 logarithms in our final expression I'll rewrite both of them in exponential form:
\n" ); document.write( "\"14%5Ea+=+7\" and \"14%5Eb+=+5\"
\n" ); document.write( "The key here is to recognize that the connection between these equations and the desired expression is that 7*5 = 35. If we take these two equations and multiply their left sides and multiply their right sides we get:
\n" ); document.write( "\"14%5Ea+%2A+14%5Eb+=+7%2A5\"
\n" ); document.write( "Using the rule for exponents on the left side we get:
\n" ); document.write( "\"14%5E%28a%2Bb%29+=+35\"
\n" ); document.write( "Now we can find the base 35 log of each side:
\n" ); document.write( "\"log%2835%2C+%2814%5E%28a%2Bb%29%29%29+=+log%2835%2C+%2835%29%29\"
\n" ); document.write( "which simplifies to:
\n" ); document.write( "\"log%2835%2C+%2814%5E%28a%2Bb%29%29%29+=+1\"
\n" ); document.write( "We have a base 35 logarithm now. Next we want to manipulate this equation to tell us what \"log%2835%2C+%2828%29%29\" would be. We'll start by using the property of logarithms, \"log%28a%2C+%28p%5Eq%29%29+=+q%2Alog%28a%2C+%28p%29%29\", to move the exponent out in front:
\n" ); document.write( "\"%28a%2Bb%29%2Alog%2835%2C+%2814%29%29+=+1\"
\n" ); document.write( "Next divide both sides by (a+b):
\n" ); document.write( "\"log%2835%2C+%2814%29%29+=+1%2F%28a%2Bb%29\"
\n" ); document.write( "We're getting pretty close. Now how do we change the log of 14 into the log of 28? Answer: with some creative factoring:
\n" ); document.write( "\"log%2835%2C+%2828%2A%281%2F2%29%29%29+=+1%2F%28a%2Bb%29\"
\n" ); document.write( "With this, we can use another property of logarithms, \"log%28a%2C+%28p%2Aq%29%29+=+log%28a%2C+%28p%29%29+%2B+log%28a%2C+%28q%29%29\", to separate the 28 and the 1/2:
\n" ); document.write( "\"log%2835%2C+%2828%29%29+%2B+log%2835%2C+%281%2F2%29%29+=+1%2F%28a%2Bb%29\"
\n" ); document.write( "Now we just subtract the log of 1/2 from each side:
\n" ); document.write( "\"log%2835%2C+%2828%29%29+=+1%2F%28a%2Bb%29+-+log%2835%2C+%281%2F2%29%29\"
\n" ); document.write( "This may be an acceptable answer. But we could simplify this a little further using yet another property of logarithms, \"log%28a%2C+%28p%2Fq%29%29+=+log%28a%2C+%28p%29%29+-+log%28a%2C+%28q%29%29\", to simplify the log of 1/2:
\n" ); document.write( "
\n" ); document.write( "Since the log of 1 is zero for any base, this simplifies as follows:
\n" ); document.write( "\"log%2835%2C+%2828%29%29+=+1%2F%28a%2Bb%29+-+%280+-+log%2835%2C+%282%29%29%29\"
\n" ); document.write( "\"log%2835%2C+%2828%29%29+=+1%2F%28a%2Bb%29+-+%28-log%2835%2C+%282%29%29%29\"
\n" ); document.write( "\"log%2835%2C+%2828%29%29+=+1%2F%28a%2Bb%29+%2B+log%2835%2C+%282%29%29\"\r
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