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Functions and Their Modeling

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:

23=82^3 = 823=8

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

log⁡28=3\log_2 8 = 3log2​8=3

In general, the relationship between exponential and logarithm:

y=bx⇔x=log⁡byy = b^x \quad \Leftrightarrow \quad x = \log_b yy=bx⇔x=logb​y

Definition and Types of Logarithms

A logarithmic function with base bbb (where b>0b > 0b>0 and b≠1b \neq 1b=1) is expressed as:

f(x)=log⁡bxfor every x>0f(x) = \log_b x \quad \text{for every } x > 0f(x)=logb​xfor every x>0

Types of logarithms commonly used:

  1. Common Logarithm (base 10): f(x)=log⁡xf(x) = \log xf(x)=logx

    Example: log⁡100=2\log 100 = 2log100=2 because 102=10010^2 = 100102=100

  2. Natural Logarithm (base e≈2.7183e \approx 2.7183e≈2.7183): f(x)=ln⁡xf(x) = \ln xf(x)=lnx

    Example: ln⁡e=1\ln e = 1lne=1 because e1=ee^1 = ee1=e

  3. Binary Logarithm (base 2): f(x)=log⁡2xf(x) = \log_2 xf(x)=log2​x

    Example: log⁡28=3\log_2 8 = 3log2​8=3 because 23=82^3 = 823=8

Logarithmic Function Graph

Comparison of Exponential and Logarithmic Functions
Graph y=log⁡2xy = \log_2 xy=log2​x is the reflection of y=2xy = 2^xy=2x across the line y=xy = xy=x.

Graph characteristics f(x)=log⁡bxf(x) = \log_b xf(x)=logb​x with b>1b > 1b>1:

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

Properties of Logarithms

Basic Properties

log⁡b1=0\log_b 1 = 0logb​1=0
log⁡bb=1\log_b b = 1logb​b=1
log⁡bbn=n\log_b b^n = nlogb​bn=n
blog⁡bx=xb^{\log_b x} = xblogb​x=x

Operational Properties

log⁡b(xy)=log⁡bx+log⁡by\log_b (xy) = \log_b x + \log_b ylogb​(xy)=logb​x+logb​y
log⁡b(xy)=log⁡bx−log⁡by\log_b \left(\frac{x}{y}\right) = \log_b x - \log_b ylogb​(yx​)=logb​x−logb​y
log⁡bxn=n⋅log⁡bx\log_b x^n = n \cdot \log_b xlogb​xn=n⋅logb​x
log⁡bx=log⁡axlog⁡ab(change of base)\log_b x = \frac{\log_a x}{\log_a b} \quad \text{(change of base)}logb​x=loga​bloga​x​(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)=2e160ln⁡(5059)tP(t) = 2e^{\frac{1}{60}\ln(5059)t}P(t)=2e601​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=2e160ln⁡(5059)t50000 = 2e^{\frac{1}{60}\ln(5059)t}50000=2e601​ln(5059)t
t=60⋅ln⁡(25000)ln⁡(5059)≈81.4 dayst = \frac{60 \cdot \ln(25000)}{\ln(5059)} \approx 81.4 \text{ days}t=ln(5059)60⋅ln(25000)​≈81.4 days

Exercises

  1. Determine the value of:

    • log⁡327\log_3 27log3​27
    • log⁡51125\log_5 \frac{1}{125}log5​1251​
    • ln⁡e3\ln e^3lne3

  2. If log⁡2x=4\log_2 x = 4log2​x=4, determine the value of xxx.

  3. Simplify: log⁡28+log⁡24−log⁡22\log_2 8 + \log_2 4 - \log_2 2log2​8+log2​4−log2​2

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

Answer Key

    • log⁡327=3\log_3 27 = 3log3​27=3
    • log⁡51125=−3\log_5 \frac{1}{125} = -3log5​1251​=−3
    • ln⁡e3=3\ln e^3 = 3lne3=3
  2. x=24=16x = 2^4 = 16x=24=16
  3. log⁡28+log⁡24−log⁡22=3+2−1=4\log_2 8 + \log_2 4 - \log_2 2 = 3 + 2 - 1 = 4log2​8+log2​4−log2​2=3+2−1=4

  4. 8000=1000⋅2t⇒8=2t⇒t=38000 = 1000 \cdot 2^t \Rightarrow 8 = 2^t \Rightarrow t = 38000=1000⋅2t⇒8=2t⇒t=3 years

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Trigonometric Function Graph

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Logarithmic Function Graph

  • Logarithmic Function ConceptMaster logarithmic functions with graphs, properties, and real-world applications. Learn the inverse of exponentials through examples and practice problems.
On this page
  • What is a Logarithmic Function?
  • Definition and Types of Logarithms
  • Logarithmic Function Graph
  • Properties of Logarithms
    • Basic Properties
    • Operational Properties
  • COVID-19 Spread Model
  • Exercises
    • Answer Key
  • Comments
  • Report
  • Source code