Question 1146418: Please help me evaluate log27 base 3 + log6 base 3 - log54 base 3 Found 2 solutions by Theo, MathTherapy:Answer by Theo(13342) (Show Source):
the rules of logarithms state that log(a*b/c) = log(a) + log(b) - log(c).
this rule applies regardless of the base of the logarithm as long as all the bases are the same.
this is the case in your problem.
also, the logarithms base conversion rule states that loga(x) = logb(x)/logb(a)
loga(x) means log(x) base a, as you have shown in your statement of the problem.
the log conversion rule is especially useful when using your calculator to find the log of any log to the base other than 10 (log function) or base other than e (ln function).
therefore, your problem can be solved as follows.
log3(27) + log3(6) - log3(54) = log3(27*6/54) = log10(27*6/54)/log10(3) = log(27*6/54)/log(3) using the log function of your calculator.
you could also have used the ln function of your calculator and gotten the same answer as shown below.
you would get:
log(27*6/54)/log(3) = 1
ln(27*6/54)/ln(3) = 1
you could also have converted to base 10 before anything else as shown below:
at this point, you could have just used your calculator to get (log(27) + log(6) - log(54)) / log(3) = 1, or you could have converted using the rules of logarithms to get log(27 * 6 / 54) / log(3) = 1.
using log function without specifying the base implies log10.
this is the default for all calculator log functions that i know of.
