DEPARTMENT OF MATHEMATICSMATH 124, Unit 2, Objective 3, Homework 2.3Compute The Following Logarithm. Round To One Decimal Place If Necessary.$\log_{4/3}\left(\frac{16}{9}\right) = \square$Submit This Problem For Grading.

Department of Mathematics MATH 124, Unit 2, Objective 3, Homework 2.3

Introduction

In this homework assignment, we will be computing a logarithm using a specific base. The problem requires us to find the value of $\log_{4/3}\left(\frac{16}{9}\right)$ and round it to one decimal place if necessary. This problem is an essential part of understanding logarithmic functions and their applications in various fields of mathematics.

Understanding Logarithms

Before we dive into the problem, let's briefly review what logarithms are. A logarithm is the inverse operation of exponentiation. In other words, it is the power to which a base number must be raised to produce a given value. For example, if we have $\log_{2}8 = 3$, it means that $2^3 = 8$. Logarithms are used to solve equations involving exponential functions and are a crucial tool in mathematics, science, and engineering.

The Problem

The problem requires us to compute the value of $\log_{4/3}\left(\frac{16}{9}\right)$. To do this, we need to use the change of base formula, which states that $\log_{b}a = \frac{\log_{c}a}{\log_{c}b}$, where $a$, $b$, and $c$ are positive real numbers and $c \neq 1$. We can choose any base for the logarithm, but in this case, we will use the natural logarithm (base $e$) for simplicity.

Step 1: Apply the Change of Base Formula

Using the change of base formula, we can rewrite the given expression as:

$$\log_{4/3}\left(\frac{16}{9}\right) = \frac{\log_{e}\left(\frac{16}{9}\right)}{\log_{e}\left(\frac{4}{3}\right)}$$

Step 2: Simplify the Expression

Now, let's simplify the expression by evaluating the logarithms in the numerator and denominator.

$$\log_{e}\left(\frac{16}{9}\right) = \log_{e}(16) - \log_{e}(9)$$

$$\log_{e}\left(\frac{4}{3}\right) = \log_{e}(4) - \log_{e}(3)$$

Substituting these values back into the original expression, we get:

$$\log_{4/3}\left(\frac{16}{9}\right) = \frac{\log_{e}(16) - \log_{e}(9)}{\log_{e}(4) - \log_{e}(3)}$$

Step 3: Evaluate the Logarithms

Now, let's evaluate the logarithms using a calculator or a logarithm table.

$$\log_{e}(16) \approx 2.7726$$

$$\log_{e}(9) \approx 2.1972$$

$$\log_{e}(4) \approx 1.3863$$

$$\log_{e}(3) \approx 1.0986$$

Substituting these values back into the expression, we get:

$$\log_{4/3}\left(\frac{16}{9}\right) \approx \frac{2.7726 - 2.1972}{1.3863 - 1.0986}$$

Step 4: Simplify the Expression

Now, let's simplify the expression by evaluating the numerator and denominator.

$$\log_{4/3}\left(\frac{16}{9}\right) \approx \frac{0.5754}{0.2877}$$

Step 5: Round to One Decimal Place

Finally, let's round the result to one decimal place.

$$\log_{4/3}\left(\frac{16}{9}\right) \approx 2.0$$

Conclusion

In this homework assignment, we computed the value of $\log_{4/3}\left(\frac{16}{9}\right)$ using the change of base formula and evaluated the logarithms using a calculator or a logarithm table. The final answer is $\boxed{2.0}$.

Additional Tips and Resources

• To practice computing logarithms, try using different bases and values.
• Use a calculator or a logarithm table to evaluate logarithms.
• Review the change of base formula and practice applying it to different problems.
• For more information on logarithms, check out the following resources:
  • Khan Academy: Logarithms
  • Mathway: Logarithms
  • Wolfram Alpha: Logarithms
    Department of Mathematics MATH 124, Unit 2, Objective 3, Homework 2.3

Q&A: Frequently Asked Questions about Logarithms

Q: What is a logarithm?

A: A logarithm is the inverse operation of exponentiation. It is the power to which a base number must be raised to produce a given value.

Q: What is the change of base formula?

A: The change of base formula is a mathematical formula that allows us to change the base of a logarithm. It is given by:

$$\log_{b}a = \frac{\log_{c}a}{\log_{c}b}$$

where $a$, $b$, and $c$ are positive real numbers and $c \neq 1$.

Q: How do I apply the change of base formula?

A: To apply the change of base formula, simply substitute the values of $a$, $b$, and $c$ into the formula and simplify.

Q: What is the natural logarithm?

A: The natural logarithm is a logarithm with a base of $e$, where $e$ is a mathematical constant approximately equal to 2.71828.

Q: How do I evaluate logarithms?

A: To evaluate logarithms, you can use a calculator or a logarithm table. Alternatively, you can use the change of base formula to change the base of the logarithm to a more manageable value.

Q: What is the difference between a logarithm and an exponent?

A: A logarithm is the inverse operation of exponentiation. In other words, it is the power to which a base number must be raised to produce a given value. An exponent, on the other hand, is the power to which a base number is raised to produce a given value.

Q: How do I use logarithms in real-world applications?

A: Logarithms have many real-world applications, including:

• Finance: Logarithms are used to calculate interest rates and investment returns.
• Science: Logarithms are used to calculate the pH of a solution and the concentration of a substance.
• Engineering: Logarithms are used to calculate the power of a signal and the gain of an amplifier.

Q: What are some common logarithmic identities?

A: Some common logarithmic identities include:

• Product rule: $\log_{b}(xy) = \log_{b}x + \log_{b}y$
• Quotient rule: $\log_{b}\left(\frac{x}{y}\right) = \log_{b}x - \log_{b}y$
• Power rule: $\log_{b}x^{n} = n\log_{b}x$

Q: How do I solve logarithmic equations?

A: To solve logarithmic equations, you can use the following steps:

1. Isolate the logarithm: Move all terms except the logarithm to one side of the equation.
2. Apply the change of base formula: Change the base of the logarithm to a more manageable value.
3. Simplify the equation: Simplify the equation by combining like terms.
4. Solve for the variable: Solve for the variable by isolating it on one side of the equation.

Conclusion

In this Q&A article, we have covered some of the most frequently asked questions about logarithms. We have discussed the change of base formula, how to apply it, and how to evaluate logarithms. We have also covered some common logarithmic identities and how to solve logarithmic equations. We hope that this article has been helpful in answering your questions about logarithms.