SOLUTION: Write as a sum or difference of individual logarithms of x, y, and z: log(a)(x^4/yz^2)

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Question 202938: Write as a sum or difference of individual logarithms of x, y, and z: log(a)(x^4/yz^2)
Found 2 solutions by jsmallt9, RAY100:
Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
There are several properties of logarithms which are useful when you want to manipulate expressions involving them:
  1. log%28a%2C+%28x%2Ay%29%29+=+log%28a%2C+%28x%29%29+%2B+log%28a%2C+%28y%29%29
    Used from left to right, this property can be used to separate factors in the argument of a logarithm into separate logarithms. Used from right to left this can be used to combine the sum of two logarithms into a single, equivalent logarithm.
  2. log%28a%2C+%28x%2Fy%29%29+=+log%28a%2C+%28x%29%29+-+log%28a%2C+%28y%29%29
    Used from left to right, this property can be used to separate the numerator and denominator of a fraction in the argument of a logarithm into separate logarithms. Used from right to left this can be used to combine the difference of two logarithms into a single, equivalent logarithm.
  3. log%28a%2C+%28x%5Ey%29%29+=+y%2Alog%28a%2C+%28x%29%29
    Used from left to right, this property can be used to "move" of the argument of a logarithm out in front of the logarithm (as a coefficient. Used from right to left this can be used to "move" a coefficient of a logarithm into the arguments as the exponent of the logarithm.
  4. log%28a%2C+%28x%29%29+=+%28log%28b%2C+%28x%29%29%29%2F%28log%28b%2C+%28a%29%29%29
    This property is used most used from left to right in order to change the base of a logarithm from "a" to "b".

Since we are interested in separating the x's, y's and z's into separate terms we will be using the first three properties from left to right.

log%28a%2C+%28%28x%5E4%29%2F%28yz%5E2%29%29%29
Since the argument is a fraction, I'll use property #2 to split the fraction into separate logs:
log%28a%2C+%28x%5E4%29%29+-+log%28a%2C+%28yz%5E2%29%29
Now I can move the exponent of the argument of the first log out in front using property #3:
4%2Alog%28a%2C+%28x%29%29+-+log%28a%2C+%28yz%5E2%29%29
Now I'll separate the product in the argument of the second log using property #1:
4%2Alog%28a%2C+%28x%29%29+-+%28log%28a%2C+%28y%29%29+%2B+log%28a%2C+%28z%5E2%29%29%29
Note the parentheses around the new expression. This is critical since there is a subtraction in front! Next I'll "move" the exponent out the argument of the 3rd log using property #3:
4%2Alog%28a%2C+%28x%29%29+-+%28log%28a%2C+%28y%29%29+%2B+2%2Alog%28a%2C+%28z%29%29%29
And finally I'll subtract the expression in the parentheses:
4%2Alog%28a%2C+%28x%29%29+-+log%28a%2C+%28y%29%29+-+2%2Alog%28a%2C+%28z%29%29

Answer by RAY100(1637) About Me  (Show Source):
You can put this solution on YOUR website!
Remember log (a/b) =log(a) -log(b),,,,and log(a*b) = log(a) + log(b),,,,,
and log(a^b) = b*log(a)
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log(a) {x^4 / yz^2},,,,,
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log(a) x^4 - log(a)y -log(a)z^2
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4*log(a)x-log(a)y -2*log(a)z
.
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check,,Let a=10,,,,x=1,,,y=2,,,z=3
original,,,log(10) {(1)^4 / (2)(3^2) } = -1.255
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answer,,,4 * log(10)1 -log(10)2 -2*log(10) 3 = 0 - .3-2*.477= -1.255,,,,ok