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Logarithmic Function Concept

Logarithmic functions are inverses of exponential functions. Graphs, properties, and worked problems show how the two are related.

Finding the Exponent That Produces a Value

A logarithm answers a specific question: which exponent produces a given value? In that sense, logarithmic and exponential functions undo each other.

Start with this exponential equation:

2^3 = 8

The exponent that turns $2$ into $8$ is $3$ . Logarithmic notation records the same relationship from the opposite direction:

\log_2 8 = 3

In general, exponential and logarithmic notation are related by:

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

Three bases appear especially often:

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

Example: $\log_2 8 = 3$ because

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

Exponential Model from Two Data Points

Indonesia's Ministry of Health, or Kemenkes, reported two confirmed COVID-19 cases on 2 March 2020. Kemenkes

On 30 April 2020, the ministry reported 10,118 cumulative confirmed cases. Kemenkes

The dates are $59 \text{ elapsed days}$ apart.

To practise logarithms, we can fit a simple exponential curve through those two reported values. Let $t$ be the number of elapsed days after 2 March and $P(t)$ the fitted case count. Since $P(0)=2$ and $P(59)=10{,}118$ , the model is:

P(t) = 2e^{\frac{1}{59}\ln(5059)t}

Two-Point Exponential Fit

A fitted curve through the two reported case counts, followed by a short extrapolation

If the same fitted rate were extended, logarithms would place the $50{,}000$ threshold at:

50000 = 2e^{\frac{1}{59}\ln(5059)t}

This calculation illustrates how the logarithmic model works. Real confirmed case counts also depend on testing, reporting, behaviour, and public-health measures. A fixed exponential rate is useful only while its assumptions still fit the situation.

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 $8 \text{ billion rupiah}$ ?

Worked Solutions

Read each logarithm as an exponent question. The required powers are:
$3^3=27\quad\Longrightarrow\quad\log_3 27=3$
Rewrite the logarithmic equation in exponential form:

$\log_2 x=4\quad\Longleftrightarrow\quad x=2^4=16$
Evaluate the three powers of $2$ before combining them:

$\log_2 8+\log_2 4-\log_2 2=3+2-1=4$
The formula measures $A$ in millions of rupiah, so $8 \text{ billion rupiah}=8000 \text{ million rupiah}$ . Substitute the target and isolate the exponential term:
$8000=1000\cdot2^t$

Exponential Function

Logarithmic Function Graph

Logarithmic Function ConceptLogarithmic functions are inverses of exponential functions. Graphs, properties, and worked problems show how the two are related.

On this page

Finding the Exponent That Produces a Value
Definition and Types of Logarithms
Logarithmic Function Graph
Properties of Logarithms
- Basic Properties
- Operational Properties
Exponential Model from Two Data Points
Exercises
- Worked Solutions

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Logarithmic Function Concept

Published: May 18, 2025. Updated: Sep 7, 2026.

e^1 = e

2^3 = 8

Substituting $x=16$ gives $\log_2 16=4$ , so the result checks out.

8=2^t

2^3=2^t\quad\Longrightarrow\quad t=3

The investment reaches the target after $3 \text{ years}$ .