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Use the properties of logarithms to expand the following logarithm as much as possible. Rewrite as a sum, difference, or product of logs.

[tex]\log \left(\frac{x^7 y^5}{z^9}\right)[/tex]


Sagot :

To expand the given logarithmic expression [tex]\(\log \left(\frac{x^7 y^5}{z^9}\right)\)[/tex], we can use the properties of logarithms. Let's go through the steps in detail.

1. Property of the logarithm with a quotient inside: The logarithm of a quotient is equal to the difference of the logarithms of the numerator and the denominator.
[tex]\[ \log \left(\frac{a}{b}\right) = \log(a) - \log(b) \][/tex]
Applying this property to the given expression, we get:
[tex]\[ \log \left(\frac{x^7 y^5}{z^9}\right) = \log(x^7 y^5) - \log(z^9) \][/tex]

2. Property of the logarithm with a product inside: The logarithm of a product is equal to the sum of the logarithms of the factors.
[tex]\[ \log(ab) = \log(a) + \log(b) \][/tex]
Applying this property to [tex]\(\log(x^7 y^5)\)[/tex], we get:
[tex]\[ \log(x^7 y^5) = \log(x^7) + \log(y^5) \][/tex]

3. Property of the logarithm with an exponent inside: The logarithm of a power is equal to the exponent times the logarithm of the base.
[tex]\[ \log(a^n) = n \log(a) \][/tex]
Applying this property to both [tex]\(\log(x^7)\)[/tex] and [tex]\(\log(y^5)\)[/tex], we get:
[tex]\[ \log(x^7) = 7 \log(x) \][/tex]
[tex]\[ \log(y^5) = 5 \log(y) \][/tex]

Similarly, applying this property to [tex]\(\log(z^9)\)[/tex], we get:
[tex]\[ \log(z^9) = 9 \log(z) \][/tex]

4. Combining all parts together: Now we combine all the parts we have found.
[tex]\[ \log(x^7 y^5) - \log(z^9) = \left(7 \log(x) + 5 \log(y)\right) - 9 \log(z) \][/tex]

So, the expanded form of the given logarithmic expression is:
[tex]\[ 7 \log(x) + 5 \log(y) - 9 \log(z) \][/tex]