Expert: Bobby Soltani - 8/5/2004

Question
4 questions have i

1) Evaluate the expression;
   2 (upper log)( lower 2) (upper 7)

2)Write the expression below as the sum or difference of simpler logarithmic quantities. (Assume all variables represent positive real numbers.)
  
   log (lower b) z^3x^5
                --------
                   y^2

3)Evaluate the expression:

        7 (upper log)(lower 7)(upper 5)

4) Write the expression below as the sum of difference of simpler logarithmic quantities. (Assume all variables represent positive real numbers.)

          log (lower 8) cubed root of yz

Sorry I cannot type the problems out on my computer in exactly the way they are written down in front of me, so i had to express them in the ways the you see above.
Regards, Dawn  

Answer
Hi Dawn,

1.  2^(log2 7)

For this problem, we use a property of logarithms that states:
x^(logx y)= y

Here we have x = 2 and y = 7 so,
2^(log2 7) = 7

2. logb (z^3*x^5/y^2)

For this one, we use the following properties:
logb (xy) = logb x + logb y and
logb (x/y) = logb x - logb y
logb x^a = a*logb x

Let's use this properties to simplify the expression.

logb (z^3*x^5/y^2)
=logb z^3 + logb x^5 - logb y^2
(Notice the z and x terms are positive because they are in the numberator and the y term is negative because it is in the denominator.)
Now, let's use property three above.
=3*logb z + 5*logb x - 2*logb y

That's it.

3. 7^(log7 5)

Same as number one.  7^(log7 5) = 5

4. log8((yz^(1/3)))     (yz)^(1/3) is the same as Cube Root

= (1/3)*log8 (yz)
= (1/3)*(log8 y + log8 z)
= (1/3)*log8 y + (1/3)*log8 z

Let me know if you have any questions.  Good luck.

Bobby