SOLUTION: Rewrite the expression as a single logarithm. 3 log3 (4x + 3) + 6 log3 (2x - 5)

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Question 473185: Rewrite the expression as a single logarithm.
3 log3 (4x + 3) + 6 log3 (2x - 5)
Found 3 solutions by Tatiana_Stebko, karaoz, Theo:
Answer by Tatiana_Stebko(1539) About Me  (Show Source):
You can put this solution on YOUR website!
3%2Alog%283%2C+%284x+%2B+3%29%29+%2B+6%2Alog%283%2C+%282x+-+5%29%29
Use formula n%2Alog%28b%2Ca%29=log%28b%2Ca%5En%29
log%283%2C%284x%2B3%29%5E3%29%2Blog%283%2C%282x-5%29%5E6%29
Use formula log%28c%2Ca%29%2Blog%28c%2Cb%29=log%28c%2C%28ab%29%29
log%283%2C%28%284x%2B3%29%5E3%282x-5%29%5E6%29%29

Answer by karaoz(32) About Me  (Show Source):
You can put this solution on YOUR website!
3 log3(4x + 3) + 6 log3(2x - 5) =
log3(4x + 3)3 + log3(2x - 5)6 =
log3(4x + 3)3(2x - 5)6

or

3 log3(4x + 3) + 6 log3(2x - 5) =
3 log3(4x + 3) + 3 log3(2x - 5)2 =
3 log3(4x + 3)(2x - 5)2

Answer by Theo(13342) About Me  (Show Source):
You can put this solution on YOUR website!
your expression is:
3+%2A+log%283%2C%284x+%2B+3%29%29+%2B+6+%2A+log%283%2C%282x+-+5%29%29
the concepts that are at work here are:
log%28x%5Ea%29+=+a%2Alog%28x%29
log%28x%2Ay%29+=+log%28x%29+%2B+log%28y%29
in your problem, the first concept needs to be applied first.
your equation of:
3+%2A+log%283%2C%284x+%2B+3%29%29+%2B+6+%2A+log%283%2C%282x+-+5%29%29 becomes:
log%283%2C%284x+%2B+3%29%5E3%29+%2B+log%283%2C%282x+-+5%29%5E6%29
the second concept needs to be applied now.
your equation of:
log%283%2C%28%284x+%2B+3%29%5E3%29%29+%2B+log%283%2C%28%282x+-+5%29%5E6%29%29 becomes:
log%283%2C%28%284x+%2B+3%29%5E3+%2A+%282x+-+5%29%5E6%29%29%29
the expression has been rewritten as a single logarithm.
this logarithm is to the base of 3.
i confirmed that the original expression is equivalent to the final expression so you can assume that the conversion was done correctly.
you can confirm yourself by assuming a value for x and solving for that value using the original equation and the final equation. the answers given by both should match each other.
you can test these equations out using the log function of the calculator, or using the ln function of the calculator.
it is not necessary to convert everything to the log of base 3.
you can also do that if you wish to do so, but it is not necessary to confirm if the conversion was done successfully.
assuming a base of 10 (log function of the calculator) or assuming a base of e (ln function of the calculator) will confirm if the transformation of the expression was done correctly.
this is because the laws of logarithms work regardless of the base used.
log(x*y) = log(x) + log(y) regardless of what the base is.
log(x^a) = a*log(x) regardless of what the base is.
if you are asked for a value based on a log to the base 3 however, then you must use log to the base 3.
you can convert from any base to any other base in logarithms by using the following formula:
log%28b%2C%28x%29%29+=+log%28c%2C%28x%29%29%2Flog%28c%2C%28b%29%29
if you have any questions about how to do this, let me know.