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Question Expand as many logarithms as possible: log⁡x4y3z2\log \frac{x^{4} y^{3}}{z^{2}}

Studdy Solution

STEP 1

Assumptions1. We are using the properties of logarithms to expand the given expression. . The base of the logarithm is not specified, so we assume it to be10.

STEP 2

We can use the quotient rule of logarithms to separate the numerator and the denominator of the fraction. The quotient rule states that log⁡ab=log⁡a−log⁡b\log \frac{a}{b} = \log a - \log b.
log⁡x4yz2=log⁡x4y−log⁡z2\log \frac{x^{4} y^{}}{z^{2}} = \log x^{4} y^{} - \log z^{2}

STEP 3

Now, we can use the product rule of logarithms to separate the terms in the numerator. The product rule states that log⁡ab=log⁡a+log⁡b\log ab = \log a + \log b.
log⁡xy3−log⁡z2=log⁡x+log⁡y3−log⁡z2\log x^{} y^{3} - \log z^{2} = \log x^{} + \log y^{3} - \log z^{2}

STEP 4

Finally, we can use the power rule of logarithms to move the exponents in front of the logarithms. The power rule states that log⁡an=nlog⁡a\log a^{n} = n \log a.
log⁡x4+log⁡y3−log⁡z2=4log⁡x+3log⁡y−2log⁡z\log x^{4} + \log y^{3} - \log z^{2} =4 \log x +3 \log y -2 \log zSo, the expanded form of the given logarithm is 4log⁡x+3log⁡y−2log⁡z4 \log x +3 \log y -2 \log z.

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