Question #fc497

1 Answer
JS
Sep 24, 2017

See below

Explanation:

a) Using the following two logarithmic properties:

  1. #log_a(xy) = log_ax + log_ay#

  2. #log_b(x^n) = n log_a(x)#

to rewrite the expressions:

#log_a(P^2Q)=log_a(P^2) + log_aQ = 9#

#log_a(P^2Q)=2log_a(P) + log_aQ = 9#

In terms of #log_aQ#:

#2log_a(P) + log_aQ = 9#

#2log_a(P)=9-log_aQ#

#log_a(P)=(9-log_aQ)/2#

I think this is what they are after.

b) Use the following logarithmic property:

#log_a(xy) = log_ax + log_ay#

to change the expressions to:

#log_a(PQ)=log_aP + log_aQ = 5#
#log_a(P^2Q)=log_a(P^2) + log_aQ = 9#

Now use the following logarithmic property to further simply the second equation:

#log_b(x^n) = n log_a(x)#. So:

#log_a(PQ)=log_aP + log_aQ = 5#
#log_a(P^2Q)=2log_a(P) + log_aQ = 9#

If we let:

#x=log_aP#

and

#y=log_aQ#

We can rewrite the equations above:

#x+y=5#
#2x+y=9#

Solving by Elimination - let's subtract the two equations to eliminate #y#:

#x+y=5#
#2x+y=9#

#-x=-4#
#x=4#

Substituting #x# back into one of the original equations:

#x+y=5#

#y=5-x = 5-4 = 1#

So:

#log_a(P)=4#

and

#log_a(Q)=1#

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