Identify the base and the argument

Oct 13, 2025 by miodeya.com 30 views

Introduction

Logarithms are a fundamental concept in mathematics, and they play a crucial role in various fields such as science, engineering, and finance. In this article, we will evaluate four different logarithmic expressions and provide a step-by-step solution to each problem.

Logarithm Definition

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 a base number must be raised to produce the input number. In other words, if $x = a^y$ , then $y = \log_a x$ .

Problem 1: $\log_{12} 144 = \square$

To evaluate this logarithm, we need to find the power to which 12 must be raised to produce 144.

Step 1: Factorize 144

We can factorize 144 as follows:

$144 = 12 \times 12$

Step 2: Write the logarithm in exponential form

Using the factorization from step 1, we can write the logarithm in exponential form:

$\log_{12} 144 = \log_{12} (12 \times 12)$

Step 3: Simplify the logarithm

Using the property of logarithms that $\log_a (a \times b) = \log_a a + \log_a b$ , we can simplify the logarithm:

$\log_{12} (12 \times 12) = \log_{12} 12 + \log_{12} 12$

Step 4: Evaluate the logarithm

Since $\log_a a = 1$ , we can evaluate the logarithm:

$\log_{12} 12 + \log_{12} 12 = 1 + 1 = 2$

Therefore, $\log_{12} 144 = 2$ .

Problem 2: $\log_{15} 1 = \square$

To evaluate this logarithm, we need to find the power to which 15 must be raised to produce 1.

Step 1: Recall the definition of logarithm

By definition, $\log_a 1 = 0$ for any base $a$ .

Therefore, $\log_{15} 1 = 0$ .

Problem 3: $\log_3\left(\frac{1}{81}\right) = \square$

To evaluate this logarithm, we need to find the power to which 3 must be raised to produce $\frac{1}{81}$ .

Step 1: Simplify the fraction

We can simplify the fraction as follows:

$\frac{1}{81} = \frac{1}{3^4}$

Step 2: Write the logarithm in exponential form

Using the simplification from step 1, we can write the logarithm in exponential form:

$\log_3\left(\frac{1}{81}\right) = \log_3\left(\frac{1}{3^4}\right)$

Step 3: Simplify the logarithm

Using the property of logarithms that $\log_a\left(\frac{1}{b}\right) = -\log_a b$ , we can simplify the logarithm:

$\log_3\left(\frac{1}{3^4}\right) = -\log_3 3^4$

Step 4: Evaluate the logarithm

Using the property of logarithms that $\log_a a^b = b$ , we can evaluate the logarithm:

$-\log_3 3^4 = -4$

Therefore, $\log_3\left(\frac{1}{81}\right) = -4$ .

Problem 4: $\log 0.00001 = \square$

To evaluate this logarithm, we need to find the power to which 10 must be raised to produce 0.00001.

Step 1: Convert the decimal to scientific notation

We can convert the decimal to scientific notation as follows:

$0.00001 = 1 \times 10^{-5}$

Step 2: Write the logarithm in exponential form

Using the scientific notation from step 1, we can write the logarithm in exponential form:

$\log 0.00001 = \log (1 \times 10^{-5})$

Step 3: Simplify the logarithm

Using the property of logarithms that $\log (a \times b) = \log a + \log b$ , we can simplify the logarithm:

$\log (1 \times 10^{-5}) = \log 1 + \log 10^{-5}$

Step 4: Evaluate the logarithm

Since $\log 1 = 0$ and $\log 10^{-5} = -5$ , we can evaluate the logarithm:

$\log 1 + \log 10^{-5} = 0 + (-5) = -5$

Therefore, $\log 0.00001 = -5$ .

Conclusion

Introduction

Logarithms can be a challenging concept to understand, especially for those who are new to mathematics. In this article, we will answer some of the most frequently asked questions about logarithms, covering topics such as the definition of logarithm, properties of logarithms, and how to evaluate logarithmic expressions.

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

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

A: A logarithm and an exponent are inverse operations. While an exponent raises a base number to a power, a logarithm finds the power to which a base number must be raised to produce a given number.

Q: What are the properties of logarithms?

A: There are several properties of logarithms, including:

Product property: $\log_a (b \times c) = \log_a b + \log_a c$
Quotient property: $\log_a \left(\frac{b}{c}\right) = \log_a b - \log_a c$
Power property: $\log_a b^c = c \times \log_a b$
Change of base property: $\log_a b = \frac{\log_c b}{\log_c a}$

Q: How do I evaluate a logarithmic expression?

A: To evaluate a logarithmic expression, you need to follow these steps:

Identify the base and the argument: Identify the base and the argument of the logarithm.
Use the properties of logarithms: Use the properties of logarithms to simplify the expression.
Evaluate the logarithm: Evaluate the logarithm using the properties of logarithms.

Q: What is the logarithm of 1?

A: The logarithm of 1 is 0, regardless of the base. This is because any number raised to the power of 0 is equal to 1.

Q: What is the logarithm of 0?

A: The logarithm of 0 is undefined, because any number raised to a negative power is undefined.

Q: How do I convert a logarithmic expression to exponential form?

A: To convert a logarithmic expression to exponential form, you need to follow these steps:

Identify the base and the argument: Identify the base and the argument of the logarithm.
Use the definition of logarithm: Use the definition of logarithm to rewrite the expression in exponential form.

Q: What is the difference between a common logarithm and a natural logarithm?

A: A common logarithm is a logarithm with a base of 10, while a natural logarithm is a logarithm with a base of e (approximately 2.718).

Q: How do I use a calculator to evaluate a logarithmic expression?

A: To use a calculator to evaluate a logarithmic expression, you need to follow these steps:

Enter the base: Enter the base of the logarithm.
Enter the argument: Enter the argument of the logarithm.
Press the logarithm button: Press the logarithm button to evaluate the expression.

Conclusion

In this article, we answered some of the most frequently asked questions about logarithms, covering topics such as the definition of logarithm, properties of logarithms, and how to evaluate logarithmic expressions. We hope that this article has provided a comprehensive guide to logarithms and has helped readers to understand the concept of logarithms better.