Definition

Understanding the Definition of Logarithms

Logarithms are a fundamental concept in mathematics, particularly in algebra, calculus, and complex analysis. They are the inverse operation of exponentiation and have numerous applications in science, engineering, and finance.

Definition of Logarithm

The logarithm of a number is the exponent by which another fixed number, the base, has to be raised to produce that number. The logarithm of a number x with respect to base b is denoted as log_b(x) and is defined by the following relation:

$$ b^{(\log_b(x))} = x $$

where:

  • b is the base of the logarithm (must be a positive real number not equal to 1)
  • x is the number whose logarithm is being taken (must be a positive real number)
  • log_b(x) is the logarithm of x with base b

Properties of Logarithms

Logarithms have several important properties that are useful in simplifying expressions and solving equations. Here are some of the key properties:

  1. Product Rule: The logarithm of a product is the sum of the logarithms of the factors.

$$ \log_b(m \cdot n) = \log_b(m) + \log_b(n) $$

  1. Quotient Rule: The logarithm of a quotient is the difference of the logarithms of the numerator and the denominator.

$$ \log_b\left(\frac{m}{n}\right) = \log_b(m) - \log_b(n) $$

  1. Power Rule: The logarithm of a number raised to an exponent is the exponent times the logarithm of the number.

$$ \log_b(m^n) = n \cdot \log_b(m) $$

  1. Change of Base Formula: The logarithm to one base can be converted to another base by using the following formula.

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

where k is any positive real number not equal to 1.

Types of Logarithms

There are two special types of logarithms that have their own notation and are widely used in mathematics:

Type Notation Base Definition
Common Logarithm $\log(x)$ 10 $\log_{10}(x)$
Natural Logarithm $\ln(x)$ $e$ $\log_{e}(x)$ where $e \approx 2.71828$

Examples

Let's look at some examples to understand how logarithms work:

  1. Calculating a Simple Logarithm:

Find $\log_2(8)$.

Since $2^3 = 8$, we can say $\log_2(8) = 3$.

  1. Using the Product Rule:

Simplify $\log_3(9) + \log_3(27)$.

We know that $9 = 3^2$ and $27 = 3^3$, so:

$$ \log_3(9) + \log_3(27) = \log_3(3^2) + \log_3(3^3) = 2 + 3 = 5 $$

  1. Applying the Change of Base Formula:

Convert $\log_4(16)$ to a logarithm with base 2.

Using the change of base formula:

$$ \log_4(16) = \frac{\log_2(16)}{\log_2(4)} = \frac{4}{2} = 2 $$

Since $2^2 = 4$ and $2^4 = 16$, the result is consistent.

  1. Solving a Logarithmic Equation:

Solve for x: $\log_5(x) = 3$.

We rewrite the equation in exponential form:

$$ 5^3 = x $$

Therefore, $x = 125$.

Understanding logarithms is crucial for solving various mathematical problems. They are used to simplify complex calculations, especially those involving exponential growth or decay, and are also essential in the study of sequences, series, and calculus.