Expand The Expression: $\log_{20} \frac{m}{n^5 P}$A. $\log_{20} M + 5\log_{20} N - \log_{20} P$B. $\log_{20} M - 5\log_{20} N + \log_{20} P$C. $\log_{20} M - 5\log_{20} N - \log_{20} P$D. $\log_{20} M -

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Introduction

Logarithmic expressions are a fundamental concept in mathematics, and understanding how to expand them is crucial for solving various mathematical problems. In this article, we will focus on expanding the expression log⁡20mn5p\log_{20} \frac{m}{n^5 p} using the properties of logarithms.

What are Logarithms?

Before we dive into expanding the expression, let's briefly review what logarithms are. A logarithm is the inverse operation of exponentiation. In other words, if ab=ca^b = c, then log⁡ac=b\log_a c = b. Logarithms are used to solve equations involving exponents and are a fundamental concept in mathematics.

Properties of Logarithms

There are several properties of logarithms that we will use to expand the expression. These properties are:

  • Product Property: log⁡a(xy)=log⁡ax+log⁡ay\log_a (xy) = \log_a x + \log_a y
  • Quotient Property: log⁡axy=log⁡ax−log⁡ay\log_a \frac{x}{y} = \log_a x - \log_a y
  • Power Property: log⁡axb=blog⁡ax\log_a x^b = b\log_a x

Expanding the Expression

Now that we have reviewed the properties of logarithms, let's expand the expression log⁡20mn5p\log_{20} \frac{m}{n^5 p}.

Using the quotient property, we can rewrite the expression as:

log⁡20mn5p=log⁡20m−log⁡20(n5p)\log_{20} \frac{m}{n^5 p} = \log_{20} m - \log_{20} (n^5 p)

Next, we can use the product property to rewrite the expression as:

log⁡20m−log⁡20(n5p)=log⁡20m−(log⁡20n5+log⁡20p)\log_{20} m - \log_{20} (n^5 p) = \log_{20} m - (\log_{20} n^5 + \log_{20} p)

Now, we can use the power property to rewrite the expression as:

log⁡20m−(log⁡20n5+log⁡20p)=log⁡20m−5log⁡20n−log⁡20p\log_{20} m - (\log_{20} n^5 + \log_{20} p) = \log_{20} m - 5\log_{20} n - \log_{20} p

Therefore, the expanded expression is:

log⁡20m−5log⁡20n−log⁡20p\log_{20} m - 5\log_{20} n - \log_{20} p

Conclusion

In this article, we have expanded the expression log⁡20mn5p\log_{20} \frac{m}{n^5 p} using the properties of logarithms. We have used the quotient property, product property, and power property to simplify the expression and arrive at the final answer.

Answer

The correct answer is:

C. log⁡20m−5log⁡20n−log⁡20p\log_{20} m - 5\log_{20} n - \log_{20} p

Practice Problems

Here are some practice problems to help you reinforce your understanding of expanding logarithmic expressions:

  1. Expand the expression log⁡10xy3z\log_{10} \frac{x}{y^3 z}.
  2. Expand the expression log⁡5ab2c\log_{5} \frac{a}{b^2 c}.
  3. Expand the expression log⁡2mn4p\log_{2} \frac{m}{n^4 p}.

Solutions

  1. log⁡10xy3z=log⁡10x−log⁡10(y3z)=log⁡10x−3log⁡10y−log⁡10z\log_{10} \frac{x}{y^3 z} = \log_{10} x - \log_{10} (y^3 z) = \log_{10} x - 3\log_{10} y - \log_{10} z
  2. log⁡5ab2c=log⁡5a−log⁡5(b2c)=log⁡5a−2log⁡5b−log⁡5c\log_{5} \frac{a}{b^2 c} = \log_{5} a - \log_{5} (b^2 c) = \log_{5} a - 2\log_{5} b - \log_{5} c
  3. log⁡2mn4p=log⁡2m−log⁡2(n4p)=log⁡2m−4log⁡2n−log⁡2p\log_{2} \frac{m}{n^4 p} = \log_{2} m - \log_{2} (n^4 p) = \log_{2} m - 4\log_{2} n - \log_{2} p

Final Thoughts

Introduction

In our previous article, we explored the concept of expanding logarithmic expressions using the properties of logarithms. In this article, we will delve deeper into the world of logarithmic expressions and answer some frequently asked questions.

Q&A

Q: What is the difference between a logarithmic expression and an exponential expression?

A: A logarithmic expression is the inverse operation of an exponential expression. In other words, if ab=ca^b = c, then log⁡ac=b\log_a c = b. Exponential expressions involve raising a number to a power, while logarithmic expressions involve finding the power to which a number must be raised to obtain a given value.

Q: How do I expand a logarithmic expression with multiple terms in the numerator and denominator?

A: To expand a logarithmic expression with multiple terms in the numerator and denominator, you can use the quotient property and the product property. For example, if you have the expression log⁡20mn5p\log_{20} \frac{m}{n^5 p}, you can first use the quotient property to rewrite it as log⁡20m−log⁡20(n5p)\log_{20} m - \log_{20} (n^5 p). Then, you can use the product property to rewrite it as log⁡20m−(log⁡20n5+log⁡20p)\log_{20} m - (\log_{20} n^5 + \log_{20} p).

Q: Can I use the power property to expand a logarithmic expression with a negative exponent?

A: Yes, you can use the power property to expand a logarithmic expression with a negative exponent. For example, if you have the expression log⁡201n5\log_{20} \frac{1}{n^5}, you can use the power property to rewrite it as −5log⁡20n-5\log_{20} n.

Q: How do I simplify a logarithmic expression with a coefficient in front of the logarithm?

A: To simplify a logarithmic expression with a coefficient in front of the logarithm, you can use the power property. For example, if you have the expression 2log⁡20m2\log_{20} m, you can use the power property to rewrite it as log⁡20m2\log_{20} m^2.

Q: Can I use the logarithmic properties to expand an expression with a variable in the exponent?

A: Yes, you can use the logarithmic properties to expand an expression with a variable in the exponent. For example, if you have the expression log⁡20xy\log_{20} x^y, you can use the power property to rewrite it as ylog⁡20xy\log_{20} x.

Q: How do I evaluate a logarithmic expression with a base that is not a power of 10?

A: To evaluate a logarithmic expression with a base that is not a power of 10, you can use the change of base formula. For example, if you have the expression log⁡5x\log_5 x, you can use the change of base formula to rewrite it as log⁡xlog⁡5\frac{\log x}{\log 5}.

Q: Can I use the logarithmic properties to expand an expression with a logarithm in the denominator?

A: Yes, you can use the logarithmic properties to expand an expression with a logarithm in the denominator. For example, if you have the expression log⁡xlog⁡y\frac{\log x}{\log y}, you can use the quotient property to rewrite it as log⁡yx\log_y x.

Conclusion

In this article, we have answered some frequently asked questions about logarithmic expressions. We have explored the properties of logarithms and how to use them to expand and simplify logarithmic expressions. By following the steps outlined in this article, you can become more confident in your ability to work with logarithmic expressions.

Practice Problems

Here are some practice problems to help you reinforce your understanding of logarithmic expressions:

  1. Expand the expression log⁡10xy3z\log_{10} \frac{x}{y^3 z}.
  2. Expand the expression log⁡5ab2c\log_{5} \frac{a}{b^2 c}.
  3. Expand the expression log⁡2mn4p\log_{2} \frac{m}{n^4 p}.
  4. Simplify the expression 2log⁡20m2\log_{20} m.
  5. Evaluate the expression log⁡5x\log_5 x using the change of base formula.

Solutions

  1. log⁡10xy3z=log⁡10x−log⁡10(y3z)=log⁡10x−3log⁡10y−log⁡10z\log_{10} \frac{x}{y^3 z} = \log_{10} x - \log_{10} (y^3 z) = \log_{10} x - 3\log_{10} y - \log_{10} z
  2. log⁡5ab2c=log⁡5a−log⁡5(b2c)=log⁡5a−2log⁡5b−log⁡5c\log_{5} \frac{a}{b^2 c} = \log_{5} a - \log_{5} (b^2 c) = \log_{5} a - 2\log_{5} b - \log_{5} c
  3. log⁡2mn4p=log⁡2m−log⁡2(n4p)=log⁡2m−4log⁡2n−log⁡2p\log_{2} \frac{m}{n^4 p} = \log_{2} m - \log_{2} (n^4 p) = \log_{2} m - 4\log_{2} n - \log_{2} p
  4. 2log⁡20m=log⁡20m22\log_{20} m = \log_{20} m^2
  5. log⁡5x=log⁡xlog⁡5\log_5 x = \frac{\log x}{\log 5}