Assume that x, y, z and b are positive numbers. Use the properties of logarithms to write the expression logb 4(square root)x^7y^2 / z^4 in terms of the logarithms of x, y and z
a. 28logbx+8logby-16logbz
b. 7/4 logbx+1/2 logby-logbz
c. 7/4logb(x+y-z)
d. logb x+1/2 logby-logbz
e. 7/4 logbx + logby-logbz
I am not sure how to work this out so if you could help me to work through the question I would really appreciate it. Thank you!
What are the properties of logarithms? Do you know?
http://www.google.com/#q=properties+of+logarithms
I've just looked at the link Ms. Sue posted so now I know them and have written them down in my notebook so I remember them
I'm still struggling to figure out the answer though. Ms. Sue can you help me?
I have this same question. I've looked at the link too and can't figure it out. Miriam what did you get?
Assuming all logs are base b (since the base does not matter for this kind of stuff), we have (I think)
log ∜(x^7 y^2 / z^4)
= 1/4 log(x^7 y^2 / z^4)
= 1/4 (7logx + 2logy - 4logz)
= 7/4 logx + 1/2 logy - logz
or, (b)
To write the expression logb 4(square root)x^7y^2 / z^4 in terms of the logarithms of x, y, and z, we can use the following properties of logarithms:
1. logb (ab) = logb a + logb b
2. logb (a/b) = logb a - logb b
3. logb (a^n) = n logb a
Let's break down the expression step by step:
1. Start with logb 4(square root)x^7y^2 / z^4.
2. Apply the property of the square root, which can be written as an exponent of 1/2. Therefore, the expression becomes logb 4x^(7/2)y^2 / z^4.
3. Apply the properties of logarithms to separate the terms. Using property (1), we can rewrite the expression as logb 4 + logb x^(7/2) + logb y^2 - logb z^4.
4. Simplify further using property (3). We know that logb x^(7/2) can be written as (7/2) logb x, and logb y^2 can be written as 2 logb y. The expression becomes logb 4 + (7/2) logb x + 2 logb y - logb z^4.
5. Finally, rearrange the terms using property (2). The expression becomes logb 4 + logb x^(7/2) + logb y^2 - logb z^4 = logb (4x^(7/2)y^2 / z^4).
So, the expression logb 4(square root)x^7y^2 / z^4 can be written as option (d): logb x + (1/2) logb y - logb z.
I hope this explanation helps! Let me know if you have any further questions.