SOLUTION: Hi, I've spent way to long trying to figure out this question, and could really use your help. (3/log[2](a))-(2/log[4](a))=(1/(log[1/2](a) I have to show that that statement

Algebra ->  Logarithm Solvers, Trainers and Word Problems -> SOLUTION: Hi, I've spent way to long trying to figure out this question, and could really use your help. (3/log[2](a))-(2/log[4](a))=(1/(log[1/2](a) I have to show that that statement      Log On


   

Question 537725: Hi,
I've spent way to long trying to figure out this question, and could really use your help.
(3/log[2](a))-(2/log[4](a))=(1/(log[1/2](a)
I have to show that that statement is true
I tried changing all bases to [1/2], then multiplying each denominator by the other fraction's numerator and denominator. I'm left with multiplications of logarithms minus multiplications of logarithms, and don't know what to do from there, heck, if doing that was the right thing to do in the first place!

Found 2 solutions by fcabanski, solver91311:
Answer by fcabanski(1391) About Me  (Show Source):
You can put this solution on YOUR website!
%283%2Flog%282%2Ca%29%29-%282%2Flog%284%2Ca%29%29=1%2Flog%281%2F2%2Ca%29


Step 1 is convert all the logs to the same base. Use log base 10 because if you were doing this for a final calculation, you'd use log base 10. Most calculators have a log (base 10) button.


log%282%2Ca%29=log%28a%29%2Flog%282%29


log%284%2Ca%29=log%28a%29%2Flog%284%29


log%281%2F2%2Ca%29=log%28a%29%2Flog%281%2F2%29



a/b/c = a*c/b


When there's a fraction in the denominator, flip it and multiply it by the numerator.


3%2F%28log%28a%29%2Flog%282%29%29=3%2A%28log%282%29%2Flog%28a%29%29


2%2F%28log%28a%29%2Flog%284%29%29=2%2A%28log%284%29%2Flog%28a%29%29


1%2F%28log%28a%29%2Flog%281%2F2%29%29=1%2A%28log%281%2F2%29%2Flog%28a%29%29


That manipulation gives



Remember that if %28a%2Fb%29+%2B+%28c%2Fb%29+=+%28d%2Fb%29 then a%2Bc=d


Drop the denominator log(a)


3%2Alog%282%29-2%2Alog%284%29=log%281%2F2%29


Power rule is x%2Alog%28y%29+=+log%28y%5Ex%29


log%282%5E3%29+-+log%284%5E2%29+=+log+%281%2F2%29


Quotient rule is log%28x%29-log%28y%29+=+log%28x%2Fy%29


log%28%282%5E3%29%2F%284%5E2%29%29+=+log%281%2F2%29


2%5E3=8 and 4%5E2=16


log%288%2F16%29=log%281%2F2%29


Divide 8 and 16 by 8.


log%281%2F2%29+=+log%281%2F2%29


As the Fonz would say, "Checkamundo!"

If you need help understanding math so you can solve these problems yourself, then one on one online tutoring is the answer ($30/hr). If you need faster solutions with guaranteed detailed answers, then go with personal problem solving ($3.50-$5.50 per problem). Contact me at fcabanski@hotmail.com



Answer by solver91311(24713) About Me  (Show Source):
You can put this solution on YOUR website!


I'm not sure what method you tried for base conversion, but considering your description of the result, you either used an incorrect method or made an algebra error in your process.

To convert from logarithm base to logarithm base use:



So, since you chose as your common base, do the conversions as above:



But since we know that:



We can derive that



Hence:



From which we get:



Similar analysis yields:



And then:



Can be written as:



Then multiply by the common denominator to get:



A true statement.

John

My calculator said it, I believe it, that settles it
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