Back to Mathematics syllabus

Logarithmic Functions

CSCA Logarithmic Functions study guide organized around the publicly available CSCA syllabus. Practice Mathematics questions on aicsca.com.

Before planning this topic, check the CSCA Exam Guide 2026 for exam dates, registration, fees, and subject requirements.

Syllabus Alignment

This study guide is organized around the publicly available CSCA syllabus for international undergraduate applicants.

Who It Is For

International students preparing for CSCA Math, Physics, Chemistry, or Chinese exams.

Related Practice

CSCA Practice · Go to questions

Past papers and worked video solutions · Timed mock exams · All subject video lessons

Practice by Topic

Jump from this tutorial to filtered practice questions for the same knowledge point.

Tutorial Content

Logarithmic Functions

1. Core Concepts

**Logarithmic Function** is the **Inverse Function** of the Exponential Function. The key lies in the constraints of the "Base" and the "Argument".

* **Definition**: The function $y = \log_a x$ ($a>0, a\neq 1$) is a logarithmic function.

* $x$ is the **Argument** (Definition Domain: $x > 0$).

* $a$ is the **Base**.

* **Inverse Relationship**: $y = \log_a x \iff x = a^y$.

**Special Logarithms**:

1. **Common Logarithm**: Base 10, denoted as $\lg x$ (or $\log x$ in some contexts, but $\lg$ in CSCA).

2. **Natural Logarithm**: Base $e$, denoted as $\ln x$.

---

2. Graphs and Properties

The graph of a logarithmic function always lies to the right of the y-axis and has a **Vertical Asymptote** at $x=0$. Its shape depends on base $a$.

| Feature | **$a > 1$ (e.g., $y=\log_2 x$)** | **$0 < a < 1$ (e.g., $y=\log_{0.5} x$)** |

| :--- | :--- | :--- |

| **Shape** | Rising Curve | Falling Curve |

| **Monotonicity** | **Increasing** on $(0, +\infty)$ | **Decreasing** on $(0, +\infty)$ |

| **Fixed Point** | Passes $(1, 0)$ | Passes $(1, 0)$ |

| **Range** | $\mathbb{R}$ | $\mathbb{R}$ |

| **Sign** | $y>0$ when $x>1$ | $y>0$ when $0<x<1$ |

**Symmetry**:

The graph of $y=\log_a x$ and the exponential function $y=a^x$ are symmetric about the line **$y=x$**.

---

3. Key Operational Rules

Assume $a>0, a\neq 1, M>0, N>0$.

**Basic Rules**:

1. **Product Rule**: $\log_a (MN) = \log_a M + \log_a N$

2. **Quotient Rule**: $\log_a (\frac{M}{N}) = \log_a M - \log_a N$

3. **Power Rule**: $\log_a M^n = n \log_a M$

**Advanced Formulas (Crucial for CSCA)**:

4. **Change of Base**: $\log_a b = \frac{\ln b}{\ln a}$

5. **Generalized Power Rule**: $\log_{a^n} b^m = \frac{m}{n} \log_a b$ (Exponent of base goes to denominator, exponent of argument goes to numerator).

6. **Chain Rule**: $\log_a b \cdot \log_b c = \log_a c$

---

4. Typical Examples

**Example 1: Monotonicity**

Compare: (1) $\log_2 3$ vs $\log_2 5$; (2) $\log_{0.5} 3$ vs $\log_{0.5} 5$.

**Solution**:

(1) Base $2 > 1$ (Increasing). Since $3 < 5$, then $\log_2 3 < \log_2 5$.

(2) Base $0.5 < 1$ (Decreasing). Since $3 < 5$, the inequality flips: $\log_{0.5} 3 > \log_{0.5} 5$.

**Example 2: Calculation Tricks**

Calculate $(\log_4 3 + \log_8 3)(\log_3 2 + \log_9 2)$.

**Solution**:

Simplify bases using $\log_{a^n} b = \frac{1}{n}\log_a b$.

Part 1: $\frac{1}{2}\log_2 3 + \frac{1}{3}\log_2 3 = \frac{5}{6}\log_2 3$.

Part 2: $\log_3 2 + \frac{1}{2}\log_3 2 = \frac{3}{2}\log_3 2$.

Product: $\frac{5}{6} \cdot \frac{3}{2} \cdot (\log_2 3 \cdot \log_3 2) = \frac{15}{12} = \frac{5}{4}$.

**Example 3: Domain**

Find domain of $f(x) = \log_{(2x-1)}(3-x)$.

**Solution**:

Conditions:

1. Argument $> 0 \Rightarrow x < 3$

2. Base $> 0 \Rightarrow x > 1/2$

3. Base $\neq 1 \Rightarrow x \neq 1$

Intersection: $(\frac{1}{2}, 1) \cup (1, 3)$.

---

5. Common Pitfalls

* **Fake Formula**: $\log_a (M+N) \neq \log_a M + \log_a N$. There is no rule for the log of a sum.

* **Domain**: Never forget the base condition $a \neq 1$ when finding domains.

* **Change of Base**: $\log_a b = \frac{\ln b}{\ln a}$. Remember: Argument on top, Base on bottom.