Logarithmic Function Concept

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

\log_b b = 1

\log_b b^n = n

b^{\log_b x} = x

Operational Properties

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

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

\log_b x^n = n \cdot \log_b x

\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 cases and in 60 days it reached 10,118 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 cases:

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

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

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 billion rupiah?

Answer Key

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

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