Logarithmic Function Concept

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:

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

In general, exponential and logarithmic notation are related by:

Definition and Types of Logarithms

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

Three bases appear especially often:

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 $2^3 = 8$

Logarithmic Function Graph

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

Each property follows from the definition of the logarithm as the inverse of the exponential function, because undoing the exponential function must undo its rules too. The basic properties concern a single value, and the operational properties concern a product, a quotient, or a power.

Reading a Single Logarithm

These identities follow straight from the definition, and each one has a matching property of exponentiation.

Operational Properties

These rules rewrite the logarithm of a product, a quotient, or a power as separate terms.

Exponential Model from Two Data Points

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

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

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:

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

This calculation uses logarithms to solve the exponential model. 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

These four problems move from reading a single logarithm to solving an exponential model. Work each one before the worked solutions, then check your answers against the steps that follow.

1. Determine the value of $\log_3 27$, $\log_5 \frac{1}{125}$, and $\ln e^3$.

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 $8 \text{ billion rupiah}$?

Worked Solutions

1. Read each logarithm as an exponent question. The required powers are:

   $$3^3=27\quad\Longrightarrow\quad\log_3 27=3$$

   $$5^{-3}=\frac{1}{125}\quad\Longrightarrow\quad\log_5\frac{1}{125}=-3$$

   $$e^3=e^3\quad\Longrightarrow\quad\ln e^3=3$$

2. Rewrite the logarithmic equation in exponential form:

   $$\log_2 x=4\quad\Longleftrightarrow\quad x=2^4=16$$

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

3. Evaluate the three powers of $2$ before combining them:

   $$\begin{aligned} &\log_2 8+\log_2 4-\log_2 2 \\ &=3+2-1 \\ &=4 \end{aligned}$$

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:

   $$8000=1000\cdot2^t$$

   $$8=2^t$$

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

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