Logarithm Definition

Understanding Logarithms

Logarithm is a mathematical operation that is the inverse of exponentiation. If we have an exponential equation $b = a^c$, then its logarithmic form is $^a\log b = c$.

Formal Definition of Logarithms

Let $a$ be a positive number where $0 < a < 1$ or $a > 1$, and $b > 0$, then:

Where:

• $a$ is the base of the logarithm
• $b$ is the number whose logarithm we are finding (numerus)
• $c$ is the result of the logarithm

We can read $^a\log b = c$ as: $a$ raised to what power equals $b$? The answer is $c$. Because $a^c = b$.

Relationship Between Exponents and Logarithms

Logarithms and exponents are related as operations that are inverses of each other. Consider the following examples:

Common Logarithm (Base 10)

Logarithm with base 10 is called the common logarithm. It is often simplified by omitting the base 10:

Applications of Logarithms in Exponential Growth

Determining Time to Reach a Specific Quantity

A bacterial colony initially consists of 2,000 bacteria that divide every 1 hour. The growth of these bacteria follows an exponential function:

where $x$ is time in hours.

Then, to determine the time needed for bacteria to reach a specific number, for example 100,000 bacteria, we need to find the value of $x$ that satisfies:

By dividing both sides by 2,000:

To find the value of $x$, we use the concept of logarithms:

This demonstrates that logarithms are very useful tools for solving exponential equations, especially when finding the exponent value that yields a specific result.