Expert: Sherry Wallin - 11/4/2009

Question
How would you combine logs with a different base?? My question is log base 3 (x) + log base 27 (x) = 8
I tried using change of base formula, but then they are both fractions, so how would I combine these??

Answer
Katharine~
    This is tricky but here goes. I am using _3 to indicate base 3
Using the following change of base formula:
log_a M = log_b M/log_b a  I'm going to choose my 'b' to be 3 because 27 = 3^3
We have log_27 x = [log_3 x]/[log_3 27] but 27= 3^3 so
log_27 x = [log_3 x]/[log_3 3^3]= [log_3 x]/3= (1/3)log_3 x
So now we want the sum: log_3 x + (1/3) log_3 x = (4/3)log_3 x
i.e., (4/3)log_3 x = 8 or log_3 x = (3/4)*8 which says that log_3 x = 6 and this says 3^6 = x or
x = 729. Check it in the original problem: log_3 729 + log_27 729 = 6 + 2 = 8 because
log_3 729 translates into 3 raised to what power is 729 and 3 raised to the 6th power is 729 and
27 raised to what power is 729 and 27 ^2 is 729.

Math Prof