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Use the properties of logarithms to expand the following expression.

[tex]\[ \log \left(\frac{x^5}{\sqrt[3]{y^2 z}}\right) \][/tex]

Each logarithm should involve only one variable and should not have any radicals or exponents. You may assume that all variables are positive.

[tex]\[ \log \left(\frac{x^5}{\sqrt[3]{y^2 z}}\right) = \][/tex]
[tex]\[ \square \][/tex]


Sagot :

To expand the expression [tex]\(\log \left(\frac{x^5}{\sqrt[3]{y^2 z}}\right)\)[/tex] using properties of logarithms, we will follow these steps:

1. Use the quotient rule of logarithms:

The quotient rule states that [tex]\(\log \left(\frac{A}{B}\right) = \log(A) - \log(B)\)[/tex]. Applying this, we get:

[tex]\[ \log \left(\frac{x^5}{\sqrt[3]{y^2 z}}\right) = \log(x^5) - \log(\sqrt[3]{y^2 z}) \][/tex]

2. Use the power rule of logarithms on [tex]\(\log(x^5)\)[/tex]:

The power rule states that [tex]\(\log(A^k) = k \log(A)\)[/tex]. Applying this rule to [tex]\(\log(x^5)\)[/tex], we get:

[tex]\[ \log(x^5) = 5 \log(x) \][/tex]

3. Simplify the denominator inside the logarithm:

Recall that [tex]\(\sqrt[3]{y^2 z}\)[/tex] is the same as [tex]\((y^2 z)^{1/3}\)[/tex]. Therefore:

[tex]\[ \log(\sqrt[3]{y^2 z}) = \log((y^2 z)^{1/3}) \][/tex]

4. Use the power rule of logarithms again:

Apply the power rule [tex]\(\log(A^k) = k \log(A)\)[/tex] to [tex]\(\log((y^2 z)^{1/3})\)[/tex], we get:

[tex]\[ \log((y^2 z)^{1/3}) = \frac{1}{3} \log(y^2 z) \][/tex]

5. Use the product rule of logarithms:

The product rule states that [tex]\(\log(AB) = \log(A) + \log(B)\)[/tex]. Applying this to [tex]\(\log(y^2 z)\)[/tex], we get:

[tex]\[ \log(y^2 z) = \log(y^2) + \log(z) \][/tex]

6. Use the power rule of logarithms on [tex]\(\log(y^2)\)[/tex]:

Applying [tex]\(\log(A^k) = k \log(A)\)[/tex] to [tex]\(\log(y^2)\)[/tex], we get:

[tex]\[ \log(y^2) = 2 \log(y) \][/tex]

Therefore:

[tex]\[ \log(y^2 z) = 2 \log(y) + \log(z) \][/tex]

7. Combine the results:

Substitute back to get:

[tex]\[ \frac{1}{3} \log(y^2 z) = \frac{1}{3} [2 \log(y) + \log(z)] = \frac{2}{3} \log(y) + \frac{1}{3} \log(z) \][/tex]

8. Substitute everything back into the original expression:

Now combine all the results from the steps above:

[tex]\[ \log \left(\frac{x^5}{\sqrt[3]{y^2 z}}\right) = 5 \log(x) - \left(\frac{2}{3} \log(y) + \frac{1}{3} \log(z)\right) \][/tex]

Therefore, we can distribute the negative sign:

[tex]\[ \log \left(\frac{x^5}{\sqrt[3]{y^2 z}}\right) = 5 \log(x) - \frac{2}{3} \log(y) - \frac{1}{3} \log(z) \][/tex]

Hence, the expanded form of the given expression is:

[tex]\[ \log \left(\frac{x^5}{\sqrt[3]{y^2 z}}\right) = 5 \log(x) - 0.666666666666667 \log(y) - 0.333333333333333 \log(z) \][/tex]
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