# Nakafa Learning Content

> For AI agents: use [llms.txt](https://nakafa.com/llms.txt) for the site index. Markdown versions are available by appending `.md` to content URLs or sending `Accept: text/markdown`.

URL: https://nakafa.com/en/subjects/mathematics/function-modeling/logarithmic-function-concept
Source: https://raw.githubusercontent.com/nakafaai/nakafa.com/refs/heads/main/packages/contents/material/lesson/mathematics/function-modeling/logarithmic-function-concept/en.mdx

Learn logarithmic functions with graphs, properties, and real-world applications. Learn the inverse of exponentials through examples and practice problems.

---

## What is a Logarithmic Function?

Have you ever wondered how long it takes for your investment to double? The answer lies in logarithmic functions! A logarithm is the "inverse" of an exponential. If exponential answers "what is the result?", then logarithm answers "what is the exponent?".

Let's start with a simple example. If we have:

```math
2^3 = 8
```

Question: "What power of $$2$$ gives $$8$$?" The answer is $$3$$. This is what logarithm answers:

Visible text: Question: "What power of gives ?" The answer is . This is what logarithm answers:

```math
\log_2 8 = 3
```

In general, the relationship between exponential and logarithm:

```math
y = b^x \quad \Leftrightarrow \quad x = \log_b y
```

## Definition and Types of Logarithms

A logarithmic function with base $$b$$ (where $$b > 0$$ and $$b \neq 1$$) is expressed as:

Visible text: A logarithmic function with base (where and ) is expressed as:

```math
f(x) = \log_b x \quad \text{for every } x > 0
```

**Types of logarithms commonly used:**

1. **Common Logarithm** (base $$10$$): $$f(x) = \log x$$

   Example: $$\log 100 = 2$$ because $$10^2 = 100$$

2. **Natural Logarithm** (base $$e \approx 2.7183$$): $$f(x) = \ln x$$

   Example: $$\ln e = 1$$ because $$e^1 = e$$

3. **Binary Logarithm** (base $$2$$): $$f(x) = \log_2 x$$

   Example: $$\log_2 8 = 3$$ because $$2^3 = 8$$

Visible text: 1. **Common Logarithm** (base ): 

 Example: because 

2. **Natural Logarithm** (base ): 

 Example: because 

3. **Binary Logarithm** (base ): 

 Example: because

## Logarithmic Function Graph

Component: LineEquation
Props:
- title: Comparison of Exponential and Logarithmic Functions
- description: Graph $$y = \log_2 x$$ is the reflection of{" "}
$$y = 2^x$$ across the line $$y = x$$.
  Visible text: Graph is the reflection of{" "}
 across the line .
- data: [
{
points: Array.from({ length: 100 }, (_, i) => {
const x = (i / 99) * 4 - 1;
return { x, y: Math.pow(2, x), z: 0 };
}),
color: getColor("SKY"),
labels: [{ text: "y = 2^x", at: 80, offset: [1.5, 1, 0] }],
showPoints: false,
},
{
points: Array.from({ length: 100 }, (_, i) => {
const x = (i / 99) * 8 + 0.1;
return { x, y: Math.log2(x), z: 0 };
}),
color: getColor("ROSE"),
labels: [{ text: "y = log₂ x", at: 80, offset: [0.5, -1, 0] }],
showPoints: false,
},
{
points: Array.from({ length: 50 }, (_, i) => {
const x = (i / 49) * 4 - 1;
return { x, y: x, z: 0 };
}),
color: getColor("PURPLE"),
labels: [{ text: "y = x", at: 40, offset: [1, 1, 0] }],
showPoints: false,
},
]
- cameraPosition: [0, 0, 15]
- showZAxis: false

**Graph characteristics** $$f(x) = \log_b x$$ with $$b > 1$$:

Visible text: **Graph characteristics** with :

- Domain: $$x > 0$$ (positive numbers only)
- Range: All real numbers
- $$x$$-intercept: $$(1, 0)$$
- Vertical asymptote: $$y$$-axis ($$x = 0$$)
- Increasing function for $$b > 1$$

Visible text: - Domain: (positive numbers only)
- Range: All real numbers
- -intercept: 
- Vertical asymptote: -axis ()
- Increasing function for

## Properties of Logarithms

### Basic Properties

Component: MathContainer
Children:

```math
\log_b 1 = 0
```

```math
\log_b b = 1
```

```math
\log_b b^n = n
```

```math
b^{\log_b x} = x
```

### Operational Properties

Component: MathContainer
Children:

```math
\log_b (xy) = \log_b x + \log_b y
```

```math
\log_b \left(\frac{x}{y}\right) = \log_b x - \log_b y
```

```math
\log_b x^n = n \cdot \log_b x
```

```math
\log_b x = \frac{\log_a x}{\log_a b} \quad \text{(change of base)}
```

## COVID-19 Spread Model

In the early pandemic, COVID-19 spread in Indonesia could be modeled with an exponential function. If on March $$2$$, $$2020$$ there were $$2 \text{ cases}$$ and in $$60 \text{ days}$$ it reached $$10{,}118 \text{ cases}$$, then:

Visible text: In the early pandemic, COVID-19 spread in Indonesia could be modeled with an exponential function. If on March , there were and in it reached , then:

Component: ContentStack
Children:

```math
P(t) = 2e^{\frac{1}{60}\ln(5059)t}
```

Component: LineEquation
Props:
- title: COVID-19 Spread Model
- description: Exponential growth of COVID-19 cases
- data: [
{
points: Array.from({ length: 61 }, (_, i) => {
const t = i;
const P = 2 * Math.exp((1 / 60) * Math.log(5059) * t);
return { x: t, y: P, z: 0 };
}),
color: getColor("RED"),
labels: [{ text: "P(t)", at: 3, offset: [2, -0.5, 0] }],
showPoints: false,
},
]
- cameraPosition: [15, 10, 15]
- showZAxis: false

**Using logarithms**, we can calculate when there will be $$50{,}000 \text{ cases}$$:

Visible text: **Using logarithms**, we can calculate when there will be :

Component: MathContainer
Children:

```math
50000 = 2e^{\frac{1}{60}\ln(5059)t}
```

```math
t = \frac{60 \cdot \ln(25000)}{\ln(5059)} \approx 81.4 \text{ days}
```

## Exercises

1. Determine the value of:

   - $$\log_3 27$$
   - $$\log_5 \frac{1}{125}$$
   - $$\ln e^3$$

2. If $$\log_2 x = 4$$, determine the value of $$x$$.

3. Simplify: $$\log_2 8 + \log_2 4 - \log_2 2$$

4. An investment grows according to the formula $$A = 1000 \cdot 2^t$$ (in million rupiah). How many years are needed for the investment to become $$8 \text{ billion rupiah}$$?

Visible text: 1. Determine the value of:

 - 
 - 
 - 

2. If , determine the value of .

3. Simplify: 

4. An investment grows according to the formula (in million rupiah). How many years are needed for the investment to become ?

### Answer Key

1. The values are:

   - $$\log_3 27 = 3$$
   - $$\log_5 \frac{1}{125} = -3$$
   - $$\ln e^3 = 3$$

2. $$x = 2^4 = 16$$

3. $$\log_2 8 + \log_2 4 - \log_2 2 = 3 + 2 - 1 = 4$$

4. $$8000 = 1000 \cdot 2^t \Rightarrow 8 = 2^t \Rightarrow t = 3 \text{ years}$$

Visible text: 1. The values are:

 - 
 - 
 - 

2. 

3. 

4.