Applying the Change of Base Formula

Oct 13, 2025 by talynt.org 102 views

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Introduction

In mathematics, logarithms are a fundamental concept that plays a crucial role in various mathematical operations. The logarithmic function is used to find the power to which a base number must be raised to produce a given value. In this article, we will evaluate the expression $\log_2 9$ and round our answer to two decimal places.

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 the base number must be raised to produce that number. For example, if we have the expression $\log_2 8$ , it means we need to find the power to which 2 must be raised to produce 8. In this case, the answer is 3, because $2^3 = 8$ .

Evaluating the Expression

To evaluate the expression $\log_2 9$ , we need to find the power to which 2 must be raised to produce 9. This can be done using a calculator or by using the change of base formula.

Using a Calculator

We can use a calculator to find the value of $\log_2 9$ . Most calculators have a built-in logarithmic function that allows us to enter the base and the number, and it will give us the result.

Using the Change of Base Formula

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 $c \neq 1$ . We can use this formula to evaluate the expression $\log_2 9$ by changing the base to a more familiar base, such as 10.

Applying the Change of Base Formula

Using the change of base formula, we can rewrite the expression $\log_2 9$ as:

\log_2 9 = \frac{\log_{10} 9}{\log_{10} 2}

Now, we can use a calculator to find the values of $\log_{10} 9$ and $\log_{10} 2$ .

Calculating the Values

Using a calculator, we get:

\log_{10} 9 \approx 0.954242509

\log_{10} 2 \approx 0.301029996

Evaluating the Expression

Now that we have the values of $\log_{10} 9$ and $\log_{10} 2$ , we can evaluate the expression $\log_2 9$ by substituting these values into the change of base formula:

\log_2 9 = \frac{\log_{10} 9}{\log_{10} 2} \approx \frac{0.954242509}{0.301029996} \approx 3.17

Rounding the Answer

We are asked to round our answer to two decimal places. Therefore, we round the value of $\log_2 9$ to 3.17.

Conclusion

In this article, we evaluated the expression $\log_2 9$ and rounded our answer to two decimal places. We used the change of base formula to change the base of the logarithm to a more familiar base, such as 10. We then used a calculator to find the values of $\log_{10} 9$ and $\log_{10} 2$ , and finally evaluated the expression by substituting these values into the change of base formula. The final answer is 3.17.

Discussion

The expression $\log_2 9$ is a simple logarithmic expression that can be evaluated using the change of base formula. However, it is essential to understand the concept of logarithms and how to apply the change of base formula to evaluate logarithmic expressions. This article provides a step-by-step guide on how to evaluate the expression $\log_2 9$ and round the answer to two decimal places.

Final Answer

The final answer is: $\boxed{3.17}$

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Introduction

In our previous article, we evaluated the expression $\log_2 9$ and rounded our answer to two decimal places. In this article, we will provide a Q&A guide to help you understand logarithmic expressions and how to evaluate them.

Q&A

Q: What is a logarithm?

A: 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 the base number must be raised to produce that number.

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 evaluate a logarithmic expression using the change of base formula?

A: To evaluate a logarithmic expression using the change of base formula, you need to follow these steps:

Rewrite the expression using the change of base formula.
Use a calculator to find the values of the logarithms.
Substitute the values into the change of base formula.
Simplify the expression to get the final answer.

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

A: A logarithmic expression is an expression that involves a logarithm, while an exponential expression is an expression that involves an exponent. For example, $\log_2 9$ is a logarithmic expression, while $2^3$ is an exponential expression.

Q: Can I use a calculator to evaluate a logarithmic expression?

A: Yes, you can use a calculator to evaluate a logarithmic expression. Most calculators have a built-in logarithmic function that allows you to enter the base and the number, and it will give you the result.

Q: What is the base of a logarithm?

A: The base of a logarithm is the number that is raised to a power to produce the given value. For example, in the expression $\log_2 9$ , the base is 2.

Q: Can I use the change of base formula to evaluate any logarithmic expression?

A: Yes, you can use the change of base formula to evaluate any logarithmic expression. However, you need to make sure that the base of the logarithm is not equal to 1.

Q: How do I round a logarithmic expression to two decimal places?

A: To round a logarithmic expression to two decimal places, you need to follow these steps:

Evaluate the expression using the change of base formula.
Round the result to two decimal places.

Conclusion

In this article, we provided a Q&A guide to help you understand logarithmic expressions and how to evaluate them. We covered topics such as the change of base formula, evaluating logarithmic expressions, and rounding logarithmic expressions to two decimal places. We hope that this guide has been helpful in understanding logarithmic expressions.

Final Answer

The final answer is: $\boxed{3.17}$