Evaluate The Following Expression Without A Calculator:$\log_5 125$A. 5 B. 12 C. 25 D. 3

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Introduction

In this article, we will evaluate the expression $\log_5 125$ without using a calculator. This involves understanding the concept of logarithms and applying mathematical properties to simplify the expression. We will explore the properties of logarithms, including the definition of a logarithm, the change of base formula, and the properties of logarithmic functions.

Understanding Logarithms

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

Evaluating $\log_5 125$

To evaluate $\log_5 125$, we need to find the exponent to which 5 must be raised to produce 125. We can start by expressing 125 as a power of 5.

Expressing 125 as a Power of 5

We can rewrite 125 as $5^3$. This is because $5^3 = 5 \times 5 \times 5 = 125$. Therefore, we can say that $\log_5 125 = 3$.

Verifying the Result

To verify our result, we can use the definition of a logarithm. If $\log_5 125 = 3$, then $5^3 = 125$. This is indeed true, as we have already shown.

Conclusion

In conclusion, we have evaluated the expression $\log_5 125$ without using a calculator. We have used the properties of logarithms, including the definition of a logarithm and the change of base formula, to simplify the expression. We have also verified our result using the definition of a logarithm.

Final Answer

The final answer to the expression $\log_5 125$ is $\boxed{3}$.

Additional Information

• The logarithm of a number to a certain base is the exponent to which the base must be raised to produce the number.
• The change of base formula allows us to change the base of a logarithm from one base to another.
• The properties of logarithmic functions include the product rule, the quotient rule, and the power rule.

Common Mistakes

• Not understanding the definition of a logarithm.
• Not using the properties of logarithms to simplify the expression.
• Not verifying the result using the definition of a logarithm.

Tips and Tricks

• Use the properties of logarithms to simplify the expression.
• Verify the result using the definition of a logarithm.
• Use a calculator to check the result, but do not rely on it to find the answer.

Real-World Applications

• Logarithms are used in many real-world applications, including finance, science, and engineering.
• Logarithmic functions are used to model population growth, chemical reactions, and other phenomena.
• Logarithms are used to solve equations and inequalities involving exponential functions.

Conclusion

In conclusion, we have evaluated the expression $\log_5 125$ without using a calculator. We have used the properties of logarithms, including the definition of a logarithm and the change of base formula, to simplify the expression. We have also verified our result using the definition of a logarithm. The final answer to the expression $\log_5 125$ is $\boxed{3}$.

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Introduction

In our previous article, we evaluated the expression $\log_5 125$ without using a calculator. We used the properties of logarithms, including the definition of a logarithm and the change of base formula, to simplify the expression. In this article, we will answer some common questions related to evaluating the expression $\log_5 125$.

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 the exponent to which a base number must be raised to produce the input number.

Q: How do I evaluate the expression $\log_5 125$?

A: To evaluate the expression $\log_5 125$, you need to find the exponent to which 5 must be raised to produce 125. You can start by expressing 125 as a power of 5.

Q: How do I express 125 as a power of 5?

A: You can rewrite 125 as $5^3$. This is because $5^3 = 5 \times 5 \times 5 = 125$.

Q: Why is $\log_5 125 = 3$?

A: Because $5^3 = 125$, therefore $\log_5 125 = 3$.

Q: Can I use a calculator to evaluate the expression $\log_5 125$?

A: Yes, you can use a calculator to evaluate the expression $\log_5 125$. However, it is not necessary to use a calculator to find the answer.

Q: What are some common mistakes to avoid when evaluating the expression $\log_5 125$?

A: Some common mistakes to avoid when evaluating the expression $\log_5 125$ include not understanding the definition of a logarithm, not using the properties of logarithms to simplify the expression, and not verifying the result using the definition of a logarithm.

Q: What are some real-world applications of logarithms?

A: Logarithms are used in many real-world applications, including finance, science, and engineering. Logarithmic functions are used to model population growth, chemical reactions, and other phenomena.

Q: Can I use the change of base formula to evaluate the expression $\log_5 125$?

A: Yes, you can use the change of base formula to evaluate the expression $\log_5 125$. The change of base formula is $\log_b a = \frac{\log_c a}{\log_c b}$, where $c$ is any positive real number.

Q: How do I use the change of base formula to evaluate the expression $\log_5 125$?

A: To use the change of base formula to evaluate the expression $\log_5 125$, you need to choose a base $c$ and then use the formula $\log_5 125 = \frac{\log_c 125}{\log_c 5}$.

Q: What is the final answer to the expression $\log_5 125$?

A: The final answer to the expression $\log_5 125$ is $\boxed{3}$.

Conclusion

In conclusion, we have answered some common questions related to evaluating the expression $\log_5 125$. We have used the properties of logarithms, including the definition of a logarithm and the change of base formula, to simplify the expression. We have also verified our result using the definition of a logarithm. The final answer to the expression $\log_5 125$ is $\boxed{3}$.

Additional Information

• The logarithm of a number to a certain base is the exponent to which the base must be raised to produce the number.
• The change of base formula allows us to change the base of a logarithm from one base to another.
• The properties of logarithmic functions include the product rule, the quotient rule, and the power rule.

Common Mistakes

• Not understanding the definition of a logarithm.
• Not using the properties of logarithms to simplify the expression.
• Not verifying the result using the definition of a logarithm.

Tips and Tricks

• Use the properties of logarithms to simplify the expression.
• Verify the result using the definition of a logarithm.
• Use a calculator to check the result, but do not rely on it to find the answer.

Real-World Applications

• Logarithms are used in many real-world applications, including finance, science, and engineering.
• Logarithmic functions are used to model population growth, chemical reactions, and other phenomena.
• Logarithms are used to solve equations and inequalities involving exponential functions.