Evaluate The Following Logarithms.1. $\log_{12} 144 = $ $\square$2. $\log_{15} 1 = $ $\square$3. $\log_3\left(\frac{1}{81}\right) = $ $\square$4. $\log 0.00001 = $

Introduction

Logarithms are a fundamental concept in mathematics, and they play a crucial role in various fields such as physics, engineering, and computer science. 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 $y = \log_a x$. The logarithm of a number is denoted by $\log_a x$, where $a$ is the base and $x$ is the input number.

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, 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 as follows:

$\log_{12} 144 = \log_{12} 12 + \log_{12} 12$

Step 4: Evaluate the logarithm

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

$\log_{12} 144 = 1 + 1 = 2$

Therefore, the value of $\log_{12} 144$ is 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: Write the logarithm in exponential form

Using the definition of logarithm, we can write the logarithm in exponential form:

$\log_{15} 1 = x$

Step 2: Simplify the exponential form

Using the property of exponentiation that $a^0 = 1$, we can simplify the exponential form as follows:

$15^x = 1$

Step 3: Evaluate the exponent

Since $15^0 = 1$, we can evaluate the exponent as follows:

$x = 0$

Therefore, the value of $\log_{15} 1$ is 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 definition of logarithm, we can write the logarithm in exponential form:

$\log_3\left(\frac{1}{81}\right) = x$

Step 3: Simplify the exponential form

Using the property of exponentiation that $a^{-n} = \frac{1}{a^n}$, we can simplify the exponential form as follows:

$3^x = 3^{-4}$

Step 4: Evaluate the exponent

Using the property of exponentiation that $a^{-n} = \frac{1}{a^n}$, we can evaluate the exponent as follows:

$x = -4$

Therefore, the value of $\log_3\left(\frac{1}{81}\right)$ is -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: Write the logarithm in exponential form

Using the definition of logarithm, we can write the logarithm in exponential form:

$\log 0.00001 = x$

Step 2: Simplify the exponential form

Using the property of exponentiation that $a^{-n} = \frac{1}{a^n}$, we can simplify the exponential form as follows:

$10^x = 10^{-5}$

Step 3: Evaluate the exponent

Using the property of exponentiation that $a^{-n} = \frac{1}{a^n}$, we can evaluate the exponent as follows:

$x = -5$

Therefore, the value of $\log 0.00001$ is -5.

Conclusion

In this article, we evaluated four different logarithmic expressions and provided a step-by-step solution to each problem. We used the definition of logarithm, the properties of exponentiation, and the properties of logarithms to simplify the expressions and evaluate the logarithms. The results of the evaluations are as follows:

• $\log_{12} 144 = 2$
• $\log_{15} 1 = 0$
• $\log_3\left(\frac{1}{81}\right) = -4$
• $\log 0.00001 = -5$

Introduction

Logarithms are a fundamental concept in mathematics, and they play a crucial role in various fields such as physics, engineering, and computer science. In this article, we will answer some frequently asked questions about logarithms and provide a comprehensive guide to understanding this concept.

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 is the inverse operation of an exponent. While an exponent tells us how many times a base number must be multiplied by itself to produce a given number, a logarithm tells us the power to which the base number must be raised to produce a given number.

Q: What are the properties of logarithms?

A: The properties of logarithms are as follows:

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

Q: How do I evaluate a logarithm?

A: To evaluate a logarithm, you need to find the power to which the base number must be raised to produce the input number. You can use the properties of logarithms to simplify the expression and evaluate the logarithm.

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, regardless of the base. This is because any number raised to a negative power is undefined.

Q: Can I use a calculator to evaluate a logarithm?

A: Yes, you can use a calculator to evaluate a logarithm. Most calculators have a built-in logarithm function that allows you to enter the base and the input number and get the result.

Q: What are some common logarithmic expressions?

A: Some common logarithmic expressions include:

• $\log_a a = 1$
• $\log_a 1 = 0$
• $\log_a a^b = b$
• $\log_a \left(\frac{1}{a}\right) = -1$

Conclusion

In this article, we answered some frequently asked questions about logarithms and provided a comprehensive guide to understanding this concept. We hope that this article has helped readers to understand the properties of logarithms and how to evaluate them. If you have any further questions, please don't hesitate to ask.

Logarithm Formulas

Here are some common logarithmic formulas:

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

Logarithm Examples

Here are some examples of logarithmic expressions:

• $\log_2 8 = 3$
• $\log_3 27 = 3$
• $\log_4 16 = 2$
• $\log_5 25 = 2$

Logarithm Practice Problems

Here are some practice problems to help you understand logarithms:

• Evaluate $\log_2 16$
• Evaluate $\log_3 9$
• Evaluate $\log_4 4$
• Evaluate $\log_5 25$

We hope that this article has helped you to understand logarithms and how to evaluate them. If you have any further questions, please don't hesitate to ask.