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Question Simplify x−3x2\frac{x^{-3}}{x^{2}}. Express log⁡x3−log⁡y2\log x^{3} - \log y^{2} as a single log⁡\log expression.

Studdy Solution

STEP 1

Assumptions
1. Simplify the expression x−3x2\frac{x^{-3}}{x^{2}} using the laws of exponents.
2. Express log⁡x3−log⁡y2\log x^{3} - \log y^{2} as a single log⁡\log expression using the properties of logarithms.

STEP 2

First, we will simplify the expression x−3x2\frac{x^{-3}}{x^{2}}. According to the laws of exponents, when dividing powers with the same base, we subtract the exponents.
xaxb=xa−b\frac{x^{a}}{x^{b}} = x^{a-b}

STEP 3

Apply the law of exponents to the given expression.
x−3x2=x−3−2\frac{x^{-3}}{x^{2}} = x^{-3-2}

STEP 4

Subtract the exponents.
x−3−2=x−5x^{-3-2} = x^{-5}

STEP 5

According to the laws of exponents, a negative exponent means that the base is on the wrong side of the fraction. To make the exponent positive, we can rewrite x−5x^{-5} as 1x5\frac{1}{x^{5}}.
x−5=1x5x^{-5} = \frac{1}{x^{5}}

STEP 6

Now we have simplified the expression x−3x2\frac{x^{-3}}{x^{2}} to 1x5\frac{1}{x^{5}}.

STEP 7

Next, we will express log⁡x3−log⁡y2\log x^{3} - \log y^{2} as a single log⁡\log expression. According to the properties of logarithms, the difference of two logs with the same base can be written as the log of a quotient.
log⁡a−log⁡b=log⁡(ab)\log a - \log b = \log \left(\frac{a}{b}\right)

STEP 8

Apply the property of logarithms to the given expression.
log⁡x3−log⁡y2=log⁡(x3y2)\log x^{3} - \log y^{2} = \log \left(\frac{x^{3}}{y^{2}}\right)

STEP 9

We have now expressed log⁡x3−log⁡y2\log x^{3} - \log y^{2} as a single log⁡\log expression log⁡(x3y2)\log \left(\frac{x^{3}}{y^{2}}\right).

STEP 10

Combine the results from STEP_6 and STEP_9 to provide the final simplified expressions.
The simplified expression for x−3x2\frac{x^{-3}}{x^{2}} is 1x5\frac{1}{x^{5}}.
The expression log⁡x3−log⁡y2\log x^{3} - \log y^{2} as a single log⁡\log expression is log⁡(x3y2)\log \left(\frac{x^{3}}{y^{2}}\right).

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