Expand The Logarithm Fully Using The Properties Of Logs. Express The Final Answer In Terms Of Log ⁡ X \log X Lo G X , Log ⁡ Y \log Y Lo G Y , And Log ⁡ Z \log Z Lo G Z . Log ⁡ Y 5 3 Z 2 X 5 \log \frac{\sqrt[3]{y^5}}{z^2 X^5} Lo G Z 2 X 5 3 Y 5 ​ ​

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Introduction

Logarithms are a fundamental concept in mathematics, and understanding their properties is crucial for solving various mathematical problems. In this article, we will explore the process of expanding logarithms using their properties. We will focus on the given expression log⁡y53z2x5\log \frac{\sqrt[3]{y^5}}{z^2 x^5} and express the final answer in terms of log⁡x\log x, log⁡y\log y, and log⁡z\log z.

Understanding Logarithmic Properties

Before we dive into expanding the given expression, it's essential to understand the properties of logarithms. The three main properties of logarithms are:

  • Product Property: log⁡(ab)=log⁡a+log⁡b\log (ab) = \log a + \log b
  • Quotient Property: log⁡(ab)=log⁡a−log⁡b\log \left(\frac{a}{b}\right) = \log a - \log b
  • Power Property: log⁡(ab)=blog⁡a\log (a^b) = b \log a

These properties will be used to expand the given expression.

Expanding the Given Expression

Now, let's focus on expanding the given expression log⁡y53z2x5\log \frac{\sqrt[3]{y^5}}{z^2 x^5} using the properties of logarithms.

Step 1: Simplify the Expression Inside the Logarithm

The first step is to simplify the expression inside the logarithm. We can rewrite y53\sqrt[3]{y^5} as y53y^{\frac{5}{3}}.

\sqrt[3]{y^5} = y^{\frac{5}{3}}

So, the expression becomes:

log⁡y53z2x5\log \frac{y^{\frac{5}{3}}}{z^2 x^5}

Step 2: Apply the Quotient Property

Next, we can apply the quotient property to separate the logarithm into two parts.

\log \frac{y^{\frac{5}{3}}}{z^2 x^5} = \log y^{\frac{5}{3}} - \log (z^2 x^5)

Step 3: Apply the Power Property

Now, we can apply the power property to simplify the first part of the expression.

\log y^{\frac{5}{3}} = \frac{5}{3} \log y

So, the expression becomes:

53log⁡y−log⁡(z2x5)\frac{5}{3} \log y - \log (z^2 x^5)

Step 4: Apply the Product Property

Next, we can apply the product property to simplify the second part of the expression.

\log (z^2 x^5) = \log z^2 + \log x^5

Step 5: Apply the Power Property Again

Now, we can apply the power property again to simplify the second part of the expression.

\log z^2 = 2 \log z

\log x^5 = 5 \log x

So, the expression becomes:

53log⁡y−(2log⁡z+5log⁡x)\frac{5}{3} \log y - (2 \log z + 5 \log x)

Step 6: Distribute the Negative Sign

Finally, we can distribute the negative sign to simplify the expression.

\frac{5}{3} \log y - 2 \log z - 5 \log x

Conclusion

In this article, we expanded the given expression log⁡y53z2x5\log \frac{\sqrt[3]{y^5}}{z^2 x^5} using the properties of logarithms. We simplified the expression inside the logarithm, applied the quotient property, power property, and product property to arrive at the final answer. The final answer is expressed in terms of log⁡x\log x, log⁡y\log y, and log⁡z\log z.

Final Answer

The final answer is:

53log⁡y−2log⁡z−5log⁡x\boxed{\frac{5}{3} \log y - 2 \log z - 5 \log x}

References

  • [1] "Logarithmic Properties" by Math Open Reference
  • [2] "Logarithms" by Khan Academy

Related Topics

Q: What are the three main properties of logarithms?

A: The three main properties of logarithms are:

  • Product Property: log⁡(ab)=log⁡a+log⁡b\log (ab) = \log a + \log b
  • Quotient Property: log⁡(ab)=log⁡a−log⁡b\log \left(\frac{a}{b}\right) = \log a - \log b
  • Power Property: log⁡(ab)=blog⁡a\log (a^b) = b \log a

Q: How do I simplify an expression inside a logarithm?

A: To simplify an expression inside a logarithm, you can rewrite it using exponent notation. For example, y53\sqrt[3]{y^5} can be rewritten as y53y^{\frac{5}{3}}.

Q: How do I apply the quotient property to separate a logarithm into two parts?

A: To apply the quotient property, you can subtract the logarithm of the denominator from the logarithm of the numerator. For example, log⁡y53z2x5=log⁡y53−log⁡(z2x5)\log \frac{y^{\frac{5}{3}}}{z^2 x^5} = \log y^{\frac{5}{3}} - \log (z^2 x^5).

Q: How do I apply the power property to simplify a logarithm?

A: To apply the power property, you can multiply the logarithm of the base by the exponent. For example, log⁡y53=53log⁡y\log y^{\frac{5}{3}} = \frac{5}{3} \log y.

Q: How do I apply the product property to simplify a logarithm?

A: To apply the product property, you can add the logarithms of the two bases. For example, log⁡(z2x5)=log⁡z2+log⁡x5\log (z^2 x^5) = \log z^2 + \log x^5.

Q: How do I distribute a negative sign in a logarithmic expression?

A: To distribute a negative sign, you can multiply each term inside the parentheses by the negative sign. For example, −(2log⁡z+5log⁡x)=−2log⁡z−5log⁡x- (2 \log z + 5 \log x) = -2 \log z - 5 \log x.

Q: What is the final answer to the given expression log⁡y53z2x5\log \frac{\sqrt[3]{y^5}}{z^2 x^5}?

A: The final answer is:

53log⁡y−2log⁡z−5log⁡x\boxed{\frac{5}{3} \log y - 2 \log z - 5 \log x}

Q: What are some related topics to expanding logarithms?

A: Some related topics to expanding logarithms include:

Q: Where can I find more information on logarithmic properties and expanding logarithms?

A: You can find more information on logarithmic properties and expanding logarithms on websites such as Math Open Reference and Khan Academy.

Conclusion

In this article, we answered some frequently asked questions about expanding logarithms. We covered topics such as logarithmic properties, simplifying expressions inside logarithms, and applying the quotient, power, and product properties. We also provided the final answer to the given expression and some related topics to expanding logarithms.