How to express this equation in terms of m and n ?

Given #log_8 3 = m# and #log_8 5 = n#, express #log_3 50# in terms of m and n.

1 Answer
Sep 4, 2017

#log_3(50)=(1+6n)/(3m)#

Explanation:

We need to know the logarithm rules:

  • #log_a(bc)=log_a(b)+log_a(c)#
  • #log_a(b^c)=clog_a(b)#
  • #log_a(b)=log_c(b)/log_c(a)#
  • #log_a(a)=1#

Then:

#log_3(50)=log_8(50)/log_8(3)=log_8(50)/m#

Splitting up #50# into its prime factorization:

#=log_8(2*5^2)/m=(log_8(2)+log_8(5^2))/m=(log_8(8^(1/3))+log_8(5^2))/m#

#=(1/3log_8(8)+2log_8(5))/m=(1/3+2n)/m=(1+6n)/(3m)#