Given $\log_3 2 \approx 0.631$ And $\log_3 7 \approx 1.771$, What Is $\log_3 14$?A. 1.118 B. 1.893 C. 2.402 D. 3.542

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**Logarithmic Calculations: A Step-by-Step Guide** =====================================================

Introduction

Logarithms are a fundamental concept in mathematics, used to solve equations and calculate various values. In this article, we will explore the concept of logarithms and provide a step-by-step guide on how to calculate logarithmic values.

What are Logarithms?

A logarithm is the inverse operation of exponentiation. It is a mathematical function that takes a number as input and returns a value that represents the power to which a base number must be raised to produce the input number.

Logarithmic Properties

There are several properties of logarithms that are essential to understand:

  • Product Rule: log(a × b) = log(a) + log(b)
  • Quotient Rule: log(a ÷ b) = log(a) - log(b)
  • Power Rule: log(a^b) = b × log(a)

Calculating Logarithmic Values

To calculate logarithmic values, we can use the following steps:

  1. Understand the Problem: Read the problem carefully and identify the base and the value for which we need to calculate the logarithm.
  2. Use Logarithmic Properties: Apply the product rule, quotient rule, or power rule to simplify the expression.
  3. Use a Calculator or Logarithmic Table: Use a calculator or logarithmic table to find the logarithmic value.

Example Problem

Given log⁡32≈0.631\log_3 2 \approx 0.631 and log⁡37≈1.771\log_3 7 \approx 1.771, what is log⁡314\log_3 14?

Step 1: Understand the Problem

We are given two logarithmic values, log⁡32≈0.631\log_3 2 \approx 0.631 and log⁡37≈1.771\log_3 7 \approx 1.771. We need to find the value of log⁡314\log_3 14.

Step 2: Use Logarithmic Properties

We can use the product rule to simplify the expression: log⁡314=log⁡3(2×7)=log⁡32+log⁡37\log_3 14 = \log_3 (2 \times 7) = \log_3 2 + \log_3 7

Step 3: Use a Calculator or Logarithmic Table

Using a calculator or logarithmic table, we can find the value of log⁡32+log⁡37≈0.631+1.771=2.402\log_3 2 + \log_3 7 \approx 0.631 + 1.771 = 2.402

Conclusion

In this article, we explored the concept of logarithms and provided a step-by-step guide on how to calculate logarithmic values. We used the product rule, quotient rule, and power rule to simplify the expression and found the value of log⁡314\log_3 14.

Q&A

Q: What is the definition of a logarithm?

A: A logarithm is the inverse operation of exponentiation. It is a mathematical function that takes a number as input and returns a value that represents the power to which a base number must be raised to produce the input number.

Q: What are the logarithmic properties?

A: There are several properties of logarithms, including the product rule, quotient rule, and power rule.

Q: How do I calculate logarithmic values?

A: To calculate logarithmic values, you can use the following steps: understand the problem, use logarithmic properties, and use a calculator or logarithmic table.

Q: What is the value of log⁡314\log_3 14?

A: The value of log⁡314\log_3 14 is approximately 2.402.

Q: What is the base of the logarithm in the problem?

A: The base of the logarithm in the problem is 3.

Q: What is the value of log⁡32\log_3 2?

A: The value of log⁡32\log_3 2 is approximately 0.631.

Q: What is the value of log⁡37\log_3 7?

A: The value of log⁡37\log_3 7 is approximately 1.771.

Q: Can I use a calculator to find the value of log⁡314\log_3 14?

A: Yes, you can use a calculator to find the value of log⁡314\log_3 14.