# Nakafa Framework: LLM

URL: /en/subject/high-school/10/mathematics/exponential-logarithm/logarithm-definition
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Output docs content for large language models.

---

export const metadata = {
  title: "Logarithm Definition",
  description: "Master logarithms as the inverse of exponentiation. Learn formal definitions, exponential relationships, and practical applications in growth problems.",
  authors: [{ name: "Nabil Akbarazzima Fatih" }],
  date: "04/04/2025",
  subject: "Exponents and Logarithms",

};

## Understanding Logarithms

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

### Formal Definition of Logarithms

Let <InlineMath math="a" /> be a positive number where <InlineMath math="0 < a < 1" /> or <InlineMath math="a > 1" />, and <InlineMath math="b > 0" />, then:

<BlockMath math="^a\log b = c \text{ if and only if } b = a^c" />

Where:

- <InlineMath math="a" /> is the base of the logarithm
- <InlineMath math="b" /> is the number whose logarithm we are finding (numerus)
- <InlineMath math="c" /> is the result of the logarithm

We can read <InlineMath math="^a\log b = c" /> as: <InlineMath math="a" /> raised to what power equals <InlineMath math="b" />? The answer is <InlineMath math="c" />. Because <InlineMath math="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:

| Exponential Form                            | Logarithmic Form                               |
| ------------------------------------------- | ---------------------------------------------- |
| <InlineMath math="2^5 = 32" />              | <InlineMath math="^2\log 32 = 5" />            |
| <InlineMath math="3^2 = 9" />               | <InlineMath math="^3\log 9 = 2" />             |
| <InlineMath math="5^{-2} = \frac{1}{25}" /> | <InlineMath math="^5\log \frac{1}{25} = -2" /> |
| <InlineMath math="7^0 = 1" />               | <InlineMath math="^7\log 1 = 0" />             |

## Common Logarithm (Base 10)

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

<BlockMath math="^{10}\log a = \log a" />

## 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:

<BlockMath math="f(x) = 2,000(2^x)" />

where <InlineMath math="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 <InlineMath math="x" /> that satisfies:

<BlockMath math="100,000 = 2,000(2^x)" />

By dividing both sides by 2,000:

<BlockMath math="50 = 2^x" />

To find the value of <InlineMath math="x" />, we use the concept of logarithms:

<BlockMath math="x = \log_2 50" />

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