Assuming x and y and z are positive ,use properties of logarithm to write expression as a sum or difference of logarithms or multiples of logarithms. 13.log x + log y 14. log x + log y?

1 Answer
Nov 21, 2016

Answer:

#log(x)+log(y) = log(xy)#

Explanation:

#log(x)+log(y) = log(x+y)# is a basic property of logarithms, along with

  • #log(x)-log(y) = log(x/y)#
  • #log(x^a) = alog(x)#
  • #log_x(x) = 1#

To derive the property used in the given question, recall that #log_a(x)# is defined as the unique value fulfilling #a^(log_a(x)) = x#.

Then #log_a(xy)# is the unique value fulfilling #a^(log_a(xy)) = xy#.

Now, taking #a# to the power of #log_a(x)+log_a(y)#, we have

#a^(log_a(x)+log_a(y)) = a^(log_a(x))a^(log_a(y))=xy#

As #log_a(x)+log_a(y)# shares a property which is unique to #log_a(xy)#, it must be that #log_a(x)+log_a(y) = log_a(xy)#.