Math / Algebra

QuestionUse the properties of logarithms to expand the following expression. $\log \sqrt[3]{x^{7} y^{2} z}$
Each logarithm should involve only one variable and should not have any radicals or exponents. You may assume that all variables are positive. $\log \sqrt[3]{x^{7} y^{2} z}=$ $\square$

Studdy Solution

STEP 1

What is this asking? We need to rewrite a single logarithm of a complicated expression as a sum of simpler logarithms, each with only one variable. Watch out! Don't forget those logarithm rules!
Also, remember how radicals relate to fractional exponents.

STEP 2

1. Rewrite the Radical
2. Expand the Logarithm
3. Simplify the Exponents

STEP 3

Alright, let's rewrite that cube root using a fractional exponent!
Remember, the cube root of something is the same as raising it to the power of $\frac{1}{3}$ .
So, we can rewrite our expression like this:
$\log \sqrt[3]{x^{7} y^{2} z} = \log \left( x^{7} y^{2} z \right)^{\frac{1}{3}}$

STEP 4

Why did we do this?
Because having an exponent makes it easier to use our logarithm power rule later on!

STEP 5

Now, let's expand that logarithm using the product rule!
Remember, the logarithm of a product is the sum of the logarithms.
So:
$\log \left( x^{7} y^{2} z \right)^{\frac{1}{3}} = \log \left( x^{7} \right)^{\frac{1}{3}} + \log \left( y^{2} \right)^{\frac{1}{3}} + \log z^{\frac{1}{3}}$

STEP 6

We're breaking down that big logarithm into smaller, more manageable pieces, each with just one variable.
This is what the problem wants!

STEP 7

Time to simplify those exponents using the power rule of logarithms!
Remember, the logarithm of something raised to a power is the same as the power multiplied by the logarithm.
$\log \left( x^{7} \right)^{\frac{1}{3}} + \log \left( y^{2} \right)^{\frac{1}{3}} + \log z^{\frac{1}{3}} = \frac{7}{3} \log x + \frac{2}{3} \log y + \frac{1}{3} \log z$

STEP 8

We multiplied the exponents to get those nice fractions in front.
And look, each logarithm now has only one variable and no radicals or exponents left inside!
We did it!

STEP 9

Our final answer is:
$\frac{7}{3} \log x + \frac{2}{3} \log y + \frac{1}{3} \log z$

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QuestionUse the properties of logarithms to expand the following expression. $\log \sqrt[3]{x^{7} y^{2} z}$
Each logarithm should involve only one variable and should not have any radicals or exponents. You may assume that all variables are positive. $\log \sqrt[3]{x^{7} y^{2} z}=$ $\square$

Studdy Solution

STEP 1

What is this asking? We need to rewrite a single logarithm of a complicated expression as a sum of simpler logarithms, each with only one variable. Watch out! Don't forget those logarithm rules!
Also, remember how radicals relate to fractional exponents.

STEP 2

1. Rewrite the Radical
2. Expand the Logarithm
3. Simplify the Exponents

STEP 3

Alright, let's rewrite that cube root using a fractional exponent!
Remember, the cube root of something is the same as raising it to the power of $\frac{1}{3}$ .
So, we can rewrite our expression like this:
$\log \sqrt[3]{x^{7} y^{2} z} = \log \left( x^{7} y^{2} z \right)^{\frac{1}{3}}$

STEP 4

Why did we do this?
Because having an exponent makes it easier to use our logarithm power rule later on!

STEP 5

Now, let's expand that logarithm using the product rule!
Remember, the logarithm of a product is the sum of the logarithms.
So:
$\log \left( x^{7} y^{2} z \right)^{\frac{1}{3}} = \log \left( x^{7} \right)^{\frac{1}{3}} + \log \left( y^{2} \right)^{\frac{1}{3}} + \log z^{\frac{1}{3}}$

STEP 6

We're breaking down that big logarithm into smaller, more manageable pieces, each with just one variable.
This is what the problem wants!

STEP 7

STEP 8

We multiplied the exponents to get those nice fractions in front.
And look, each logarithm now has only one variable and no radicals or exponents left inside!
We did it!

STEP 9

Our final answer is:
$\frac{7}{3} \log x + \frac{2}{3} \log y + \frac{1}{3} \log z$

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Studdy Solution

STEP 1

What is this asking? We need to rewrite a *single* logarithm of a complicated expression as a *sum* of simpler logarithms, each with only *one* variable. Watch out! Don't forget those logarithm rules!Also, remember how radicals relate to fractional exponents.

STEP 2

1. Rewrite the Radical2. Expand the Logarithm3. Simplify the Exponents

STEP 3

STEP 4

Why did we do this?Because having an exponent makes it easier to use our logarithm power rule later on!

STEP 5

STEP 6

We're breaking down that big logarithm into smaller, more manageable pieces, each with just one variable.This is what the problem wants!

STEP 7

STEP 8

We multiplied the exponents to get those nice fractions in front.And look, each logarithm now has only one variable and no radicals or exponents left inside!We did it!

STEP 9

Our final answer is:73log⁡x+23log⁡y+13log⁡z\frac{7}{3} \log x + \frac{2}{3} \log y + \frac{1}{3} \log z37​logx+32​logy+31​logz

Get Studdy

Studdy solves anything!

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

What is this asking? We need to rewrite a single logarithm of a complicated expression as a sum of simpler logarithms, each with only one variable. Watch out! Don't forget those logarithm rules!
Also, remember how radicals relate to fractional exponents.

1. Rewrite the Radical
2. Expand the Logarithm
3. Simplify the Exponents

Why did we do this?
Because having an exponent makes it easier to use our logarithm power rule later on!

We're breaking down that big logarithm into smaller, more manageable pieces, each with just one variable.
This is what the problem wants!

We multiplied the exponents to get those nice fractions in front.
And look, each logarithm now has only one variable and no radicals or exponents left inside!
We did it!

Our final answer is:
$\frac{7}{3} \log x + \frac{2}{3} \log y + \frac{1}{3} \log z$