Not using the change of base formula

Oct 13, 2025 by thesweetdesigns.com 6 views

Introduction

Logarithmic expressions are a fundamental concept in mathematics, and evaluating them is a crucial skill for students and professionals alike. In this article, we will focus on evaluating the expression $\log_3 10$ and provide a step-by-step guide on how to solve it.

Understanding Logarithms

Before we dive into the evaluation of the expression, let's briefly review the concept of logarithms. A logarithm is the inverse operation of exponentiation. In other words, if $x = a^y$ , then $y = \log_a x$ . The logarithm of a number to a given base is the exponent to which the base must be raised to produce that number.

Evaluating the Expression

Now that we have a basic understanding of logarithms, let's evaluate the expression $\log_3 10$ . To do this, we need to find the exponent to which 3 must be raised to produce 10.

One way to approach this problem is to use the change of base formula, which states that $\log_a b = \frac{\log_c b}{\log_c a}$ , where $c$ is any positive real number. We can choose $c$ to be 10, which is the number we are trying to find the logarithm of.

Using the change of base formula, we get:

\log_3 10 = \frac{\log_{10} 10}{\log_{10} 3}

Since $\log_{10} 10 = 1$ , we can simplify the expression to:

\log_3 10 = \frac{1}{\log_{10} 3}

Now, we need to find the value of $\log_{10} 3$ . We can use a calculator or a logarithm table to find this value.

Using a Calculator

If we use a calculator to find the value of $\log_{10} 3$ , we get:

\log_{10} 3 \approx 0.47712125

Now, we can substitute this value back into the expression:

\log_3 10 = \frac{1}{0.47712125} \approx 2.09455149

Rounding the Answer

Since we are asked to round our answer to two decimal places, we can round the value we obtained to:

\log_3 10 \approx 2.09

Conclusion

In this article, we evaluated the expression $\log_3 10$ using the change of base formula and a calculator. We found that the value of the expression is approximately 2.09, which is closest to option A. 2.10.

Common Mistakes to Avoid

When evaluating logarithmic expressions, there are several common mistakes to avoid. Here are a few:

Not using the change of base formula: The change of base formula is a powerful tool for evaluating logarithmic expressions. Make sure to use it when necessary.
Not checking the domain: Logarithmic expressions have a domain of positive real numbers. Make sure to check the domain of the expression before evaluating it.
Not rounding correctly: When rounding a value to a certain number of decimal places, make sure to round correctly. In this case, we rounded the value of $\log_3 10$ to two decimal places.

Practice Problems

Here are a few practice problems to help you evaluate logarithmic expressions:

$\log_2 8$
$\log_5 25$
$\log_{10} 100$

Solutions

Here are the solutions to the practice problems:

$\log_2 8 = \frac{\log_{10} 8}{\log_{10} 2} \approx 3.00$
$\log_5 25 = \frac{\log_{10} 25}{\log_{10} 5} \approx 2.00$
$\log_{10} 100 = \frac{\log_{10} 100}{\log_{10} 10} \approx 2.00$

Conclusion

Introduction

In our previous article, we evaluated the expression $\log_3 10$ using the change of base formula and a calculator. We found that the value of the expression is approximately 2.09, which is closest to option A. 2.10. In this article, we will provide a Q&A guide to help you evaluate logarithmic expressions.

Q: What is the change of base formula?

A: The change of base formula is a powerful tool for evaluating logarithmic expressions. It states that $\log_a b = \frac{\log_c b}{\log_c a}$ , where $c$ is any positive real number.

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

A: To use the change of base formula, you need to choose a base $c$ and then substitute the values of $a$ , $b$ , and $c$ into the formula. For example, to evaluate $\log_3 10$ , you can choose $c$ to be 10 and then substitute the values of $a = 3$ , $b = 10$ , and $c = 10$ into the formula.

Q: What is the domain of a logarithmic expression?

A: The domain of a logarithmic expression is the set of all positive real numbers. In other words, the base of the logarithm must be positive, and the argument of the logarithm must be positive.

Q: How do I round a logarithmic expression?

A: When rounding a logarithmic expression, you need to round the value of the expression to the correct number of decimal places. For example, if you are asked to round a value to two decimal places, you need to round the value to two decimal places.

Q: What are some common mistakes to avoid when evaluating logarithmic expressions?

A: Some common mistakes to avoid when evaluating logarithmic expressions include:

Not using the change of base formula
Not checking the domain of the expression
Not rounding correctly

Q: How do I evaluate a logarithmic expression with a negative base?

A: To evaluate a logarithmic expression with a negative base, you need to use the property of logarithms that states $\log_a b = -\log_{-a} b$ . This property allows you to rewrite the expression with a positive base.

Q: How do I evaluate a logarithmic expression with a fractional base?

A: To evaluate a logarithmic expression with a fractional base, you need to use the property of logarithms that states $\log_a b = \frac{\log_c b}{\log_c a}$ , where $c$ is any positive real number. This property allows you to rewrite the expression with a positive base.

Q: What are some real-world applications of logarithmic expressions?

A: Logarithmic expressions have many real-world applications, including:

Finance: Logarithmic expressions are used to calculate interest rates and investment returns.
Science: Logarithmic expressions are used to calculate the pH of a solution and the concentration of a substance.
Engineering: Logarithmic expressions are used to calculate the gain of an amplifier and the frequency response of a system.

Conclusion

In this article, we provided a Q&A guide to help you evaluate logarithmic expressions. We discussed the change of base formula, the domain of a logarithmic expression, and how to round a logarithmic expression. We also discussed common mistakes to avoid and provided examples of real-world applications of logarithmic expressions.

Practice Problems

Here are a few practice problems to help you evaluate logarithmic expressions:

$\log_2 8$
$\log_5 25$
$\log_{10} 100$

Solutions

Here are the solutions to the practice problems:

$\log_2 8 = \frac{\log_{10} 8}{\log_{10} 2} \approx 3.00$
$\log_5 25 = \frac{\log_{10} 25}{\log_{10} 5} \approx 2.00$
$\log_{10} 100 = \frac{\log_{10} 100}{\log_{10} 10} \approx 2.00$