Final answer:
To write the expression log(7)(xy²/z³) as the sum and/or difference of logarithms, apply logarithmic properties for products, quotients, and exponents, resulting in log(7)(x) + 2 · log(7)(y) - 3 · log(7)(z).
Step-by-step explanation:
The student has asked how to write the expression log(7)(xy²/z³) as the sum and/or difference of logarithms, expressing powers as factors. This is a problem related to the properties of logarithms which are part of high school mathematics curriculum.
Using the properties of logarithms:
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- The logarithm of a product of two numbers is the sum of the logarithms of those numbers. This property tells us that log(7)(xy) = log(7)(x) + log(7)(y).
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- The logarithm of a quotient is the difference of the logarithms. Hence, log(7)(x/y) = log(7)(x) - log(7)(y).
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- When a number is raised to an exponent inside a logarithm, the exponent can be brought out front as a multiplier, which is stated as log(7)(x^y) = y · log(7)(x).
Applying these rules to the given expression, we will first deal with the quotient:
log(7)(xy²/z³) = log(7)(xy²) - log(7)(z³)
Then we address the product and the exponents inside the first logarithm:
log(7)(xy²) = log(7)(x) + log(7)(y²).
And apply the rule for exponents:
log(7)(y²) = 2 · log(7)(y) and log(7)(z³) = 3 · log(7)(z)
Combining all the parts:
log(7)(xy²/z³) = log(7)(x) + 2 · log(7)(y) - 3 · log(7)(z).