Given That $\log 3 = P$, Express The Following In Terms Of $P$:$\log \frac{10}{3} - \log \frac{1}{27} + \log 81$

Introduction

In this article, we will explore the concept of logarithms and how to express complex logarithmic expressions in terms of a given variable. We will use the given equation $\log 3 = P$ as a reference point to simplify the expression $\log \frac{10}{3} - \log \frac{1}{27} + \log 81$.

Understanding Logarithms

A logarithm is the inverse operation of exponentiation. It is a mathematical function that takes a number as input and returns the power to which a base number must be raised to produce the input number. In other words, if $y = \log_b x$, then $b^y = x$.

Properties of Logarithms

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

• Product Rule: $\log (xy) = \log x + \log y$
• Quotient Rule: $\log \left(\frac{x}{y}\right) = \log x - \log y$
• Power Rule: $\log (x^y) = y \log x$

Simplifying the Expression

Using the properties of logarithms, we can simplify the given expression as follows:

$$\log \frac{10}{3} - \log \frac{1}{27} + \log 81$$

First, we can rewrite the expression using the quotient rule:

$$\log \frac{10}{3} - \log \frac{1}{27} + \log 81 = \log 10 - \log 3 - \log \frac{1}{27} + \log 81$$

Next, we can rewrite the expression using the product rule:

$$\log 10 - \log 3 - \log \frac{1}{27} + \log 81 = \log 10 - \log 3 - \log 1 + \log 27 + \log 81$$

Now, we can simplify the expression further by using the fact that $\log 1 = 0$:

$$\log 10 - \log 3 - \log 1 + \log 27 + \log 81 = \log 10 - \log 3 + \log 27 + \log 81$$

Expressing in Terms of P

Now, we can express the simplified expression in terms of $P$ using the given equation $\log 3 = P$:

$$\log 10 - \log 3 + \log 27 + \log 81 = \log 10 - P + \log 27 + \log 81$$

We can rewrite the expression using the product rule:

$$\log 10 - P + \log 27 + \log 81 = \log 10 - P + \log (27 \cdot 81)$$

Now, we can simplify the expression further by using the fact that $27 \cdot 81 = 2187$:

$$\log 10 - P + \log (27 \cdot 81) = \log 10 - P + \log 2187$$

Final Expression

Using the power rule, we can rewrite the expression as:

$$\log 10 - P + \log 2187 = \log (10 \cdot 2187) - P$$

Now, we can simplify the expression further by using the fact that $10 \cdot 2187 = 21870$:

$$\log (10 \cdot 2187) - P = \log 21870 - P$$

Therefore, the final expression in terms of $P$ is:

$$\log \frac{10}{3} - \log \frac{1}{27} + \log 81 = \log 21870 - P$$

Conclusion

In this article, we have explored the concept of logarithms and how to express complex logarithmic expressions in terms of a given variable. We have used the given equation $\log 3 = P$ as a reference point to simplify the expression $\log \frac{10}{3} - \log \frac{1}{27} + \log 81$. The final expression in terms of $P$ is $\log 21870 - P$.

References

• [1] "Logarithms". Math Open Reference. Retrieved 2023-02-20.
• [2] "Properties of Logarithms". Math Is Fun. Retrieved 2023-02-20.

Glossary

• Logarithm: A mathematical function that takes a number as input and returns the power to which a base number must be raised to produce the input number.
• Product Rule: A property of logarithms that states $\log (xy) = \log x + \log y$.
• Quotient Rule: A property of logarithms that states $\log \left(\frac{x}{y}\right) = \log x - \log y$.
• Power Rule: A property of logarithms that states $\log (x^y) = y \log x$.

Related Articles

• [1] "Introduction to Logarithms". Math Is Fun. Retrieved 2023-02-20.
• [2] "Properties of Logarithms". Math Open Reference. Retrieved 2023-02-20.
  Q&A: Expressing Logarithmic Expressions in Terms of P =====================================================

Frequently Asked Questions

Q: What is the given equation in the problem? A: The given equation is $\log 3 = P$.

Q: What is the expression that we need to simplify in terms of P? A: The expression that we need to simplify in terms of P is $\log \frac{10}{3} - \log \frac{1}{27} + \log 81$.

Q: What are the properties of logarithms that we used to simplify the expression? A: We used the following properties of logarithms to simplify the expression:

• Product Rule: $\log (xy) = \log x + \log y$
• Quotient Rule: $\log \left(\frac{x}{y}\right) = \log x - \log y$
• Power Rule: $\log (x^y) = y \log x$

Q: How did we simplify the expression using the product rule? A: We simplified the expression using the product rule by rewriting the expression as follows:

$$\log \frac{10}{3} - \log \frac{1}{27} + \log 81 = \log 10 - \log 3 - \log \frac{1}{27} + \log 81$$

Q: How did we simplify the expression using the quotient rule? A: We simplified the expression using the quotient rule by rewriting the expression as follows:

$$\log 10 - \log 3 - \log \frac{1}{27} + \log 81 = \log 10 - \log 3 - \log 1 + \log 27 + \log 81$$

Q: What is the final expression in terms of P? A: The final expression in terms of P is $\log 21870 - P$.

Q: What is the significance of the given equation in the problem? A: The given equation $\log 3 = P$ is used as a reference point to simplify the expression $\log \frac{10}{3} - \log \frac{1}{27} + \log 81$ in terms of P.

Q: What are some related concepts to logarithms? A: Some related concepts to logarithms include:

• Exponentiation: The inverse operation of logarithms.
• Properties of Exponents: Rules for simplifying expressions involving exponents.
• Logarithmic Identities: Equations that involve logarithms and can be used to simplify expressions.

Q: How can I apply the concepts learned in this article to real-world problems? A: The concepts learned in this article can be applied to real-world problems involving logarithms, such as:

• Sound Level Measurements: Logarithmic scales are used to measure sound levels in decibels.
• pH Levels: Logarithmic scales are used to measure pH levels in chemistry.
• Finance: Logarithmic scales are used to measure returns on investment in finance.

Conclusion

In this article, we have provided a Q&A section to help readers understand the concepts learned in the article. We have covered frequently asked questions, related concepts, and real-world applications of logarithms. We hope that this article has been helpful in providing a clear understanding of logarithmic expressions in terms of P.