Logarithmic Function Concept: Functions and Their Modeling - Mathematics (Grade 11)

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:

2^3 = 8

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

\log_2 8 = 3

In general, the relationship between exponential and logarithm:

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:

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

Types of logarithms commonly used:

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

Example: $\log 100 = 2$ because $10^2 = 100$
Natural Logarithm (base $e \approx 2.7183$ ): $f(x) = \ln x$

Example: $\ln e = 1$ because $e^1 = e$
Binary Logarithm (base $2$ ): $f(x) = \log_2 x$

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

Logarithmic Function Graph

Comparison of Exponential and Logarithmic Functions

Graph

y = \log_2 x

is the reflection of

y = 2^x

across the line

y = x

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

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

Properties of Logarithms

Basic Properties

\log_b 1 = 0

Operational Properties

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

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:

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

COVID-19 Spread Model

Exponential growth of COVID-19 cases

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

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

Exercises

Determine the value of:
- $\log_3 27$
- $\log_5 \frac{1}{125}$
- $\ln e^3$
If $\log_2 x = 4$ , determine the value of $x$ .
Simplify: $\log_2 8 + \log_2 4 - \log_2 2$
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 $8 \text{ billion rupiah}$ ?

Answer Key

The values are:
- $\log_3 27 = 3$
- $\log_5 \frac{1}{125} = -3$
- $\ln e^3 = 3$
$x = 2^4 = 16$
$\log_2 8 + \log_2 4 - \log_2 2 = 3 + 2 - 1 = 4$
$8000 = 1000 \cdot 2^t \Rightarrow 8 = 2^t \Rightarrow t = 3 \text{ years}$

What is a Logarithmic Function?

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

2^3 = 8

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

\log_2 8 = 3

In general, the relationship between exponential and logarithm:

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:

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

Types of logarithms commonly used:

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

Example: $\log 100 = 2$ because $10^2 = 100$
Natural Logarithm (base $e \approx 2.7183$ ): $f(x) = \ln x$

Example: $\ln e = 1$ because $e^1 = e$
Binary Logarithm (base $2$ ): $f(x) = \log_2 x$

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

Logarithmic Function Graph

Comparison of Exponential and Logarithmic Functions

Graph

y = \log_2 x

is the reflection of

y = 2^x

across the line

y = x

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

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

Properties of Logarithms

Basic Properties

\log_b 1 = 0

Operational Properties

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

COVID-19 Spread Model

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

COVID-19 Spread Model

Exponential growth of COVID-19 cases

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

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

Exercises

Determine the value of:
- $\log_3 27$
- $\log_5 \frac{1}{125}$
- $\ln e^3$
If $\log_2 x = 4$ , determine the value of $x$ .
Simplify: $\log_2 8 + \log_2 4 - \log_2 2$
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 $8 \text{ billion rupiah}$ ?

Answer Key

The values are:
- $\log_3 27 = 3$
- $\log_5 \frac{1}{125} = -3$
- $\ln e^3 = 3$
$x = 2^4 = 16$
$\log_2 8 + \log_2 4 - \log_2 2 = 3 + 2 - 1 = 4$
$8000 = 1000 \cdot 2^t \Rightarrow 8 = 2^t \Rightarrow t = 3 \text{ years}$