Lesson 4.2: Logarithms and the Laws of Logs

Introduction

In this lesson, we will explore the fascinating world of logarithms, which serve as the inverse function to exponentials. Understanding logarithms is crucial for modeling real-world phenomena such as population growth and radioactive decay. By the end of this lesson, you will be able to convert between logarithmic and exponential forms, apply the laws of logarithms, and utilize logarithms in problem-solving.

Learning Objectives

Understand logarithms as the inverse of exponentials, including common and natural logarithms.
Familiarize yourself with the laws of logarithms and the change-of-base idea.
Convert between exponential and logarithmic form.
Apply the laws of logarithms to simplify expressions.

What is a Logarithm?

A logarithm answers the question: to what exponent must a base be raised to produce a given number? We can define the logarithm of $x$ with base $b$ as follows:

$$\log_b(x) = y \quad \text{if and only if} \quad b^y = x$$

where:

$b$ is the base (must be a positive number and not equal to 1)
$x$ is the number being analyzed (must be positive)
$y$ is the logarithm value

Common and Natural Logarithms

The two most frequently used bases in logarithms are base 10 and base $e$ (approximately 2.71828).

Common Logarithms: The logarithm with base 10 is called the common logarithm. It is typically written as $\log(x)$ without explicitly indicating the base:

$$\log_{10}(x) = \log(x)$$

Natural Logarithms: The logarithm with base $e$ is called the natural logarithm and is denoted as $\ln(x)$:

$$\log_e(x) = \ln(x)$$

Converting Between Forms

To convert between exponential and logarithmic forms, we can use the definition presented earlier. Let’s look at an example.

Example 1: Convert the exponential equation $2^3 = 8$ into logarithmic form.

Solution: Using the definition of a logarithm:

$$2^3 = 8 \implies \log_2(8) = 3$$

Conversely, if we start with the logarithmic form, we can convert it back into exponential form.

Example 2: Convert $\log_5(25) = 2$ into exponential form.

Solution:

$$\log_5(25) = 2 \implies 5^2 = 25$$

The Laws of Logarithms

Logarithms have several important properties known as the laws of logarithms. Understanding these laws enables us to simplify complex logarithmic expressions.

Product Law:

$\log_b(xy) = \log_b(x) + \log_b(y)$

This law states that the logarithm of a product is the sum of the logarithms.

Example 3: Simplify $\log_2(8 \cdot 4)$.

Solution:

$\log_2(8 \cdot 4) = \log_2(8) + \log_2(4) = 3 + 2 = 5$

Quotient Law:

$$\log_b\left(\frac{x}{y}

ight) = $\log$_b(x) - $\log$_b(y)$$

The logarithm of a quotient is the difference of the logarithms.

Example 4: Simplify $\log_3(9/3)$.

Solution:

$$\log_3\left(\frac{9}{3}

ight) = $\log_3$(9) - $\log_3$(3) = 2 - 1 = 1$$

Power Law:

$\log_b(x^p) = p \cdot \log_b(x)$

This law states that the logarithm of a number raised to a power is the power times the logarithm of the number.

Example 5: Simplify $\log_5(25^3)$.

Solution:

$\log_5(25^3) = 3 \cdot \log_5(25) = 3 \cdot 2 = 6$

Change of Base Formula

Sometimes we need to compute logarithms with a base that is not easily calculable. The change of base formula allows us to convert any logarithm to a different base conveniently. The formula is given by:

$$\log_b(x) = \frac{\log_k(x)}{\log_k(b)}$$

for any base $k$. Common choices for $k$ are 10 and $e$.

Example 6: Calculate $\log_2(16)$ using the change of base formula with base 10:

Solution:

$$\log_2(16) = \frac{\log_{10}(16)}{\log_{10}(2)}$$

We can compute this using a calculator:

$\log_{10}(16) \approx 1.2041$
$\log_{10}(2) \approx 0.3010$

Thus:

$$\log_2(16) \approx \frac{1.2041}{0.3010} \approx 4$$

Conclusion

In this lesson, we have explored logarithms as the inverse of exponentials, learned their laws, and practiced converting between forms and applying these laws. Logarithms are useful not only in mathematics but also in real-world applications involving growth and decay. Mastery of these concepts will enhance your understanding of functions and exponential growth.

Study Notes

Logarithms answer the question of what exponent is needed to produce a certain number.
The common logarithm has base 10, while the natural logarithm has base $e$.
Remember the laws of logarithms: Product, Quotient, and Power.
Use the change of base formula to compute logarithms with different bases.