对数函数duìshù hánshù

Core Concept

A Logarithmic Function is the inverse of an exponential function, with the form $y = \log_a x$ ($a > 0, a \neq 1$).

Definition

If $a^y = x$ ($a > 0, a \neq 1$), then $y$ is called the logarithm of $x$ to base $a$: $y = \log_a x$

Where:

• $a$ is the base
• $x$ is the argument
• $y$ is the logarithm

Properties

1. Domain and Range

• Domain: $(0, +\infty)$
• Range: $\mathbb{R}$

2. Fixed Point

All logarithmic functions pass through $(1, 0)$: $\log_a 1 = 0$

3. Monotonicity

• When $a > 1$: $y = \log_a x$ is increasing on $(0, +\infty)$
• When $0 < a < 1$: $y = \log_a x$ is decreasing on $(0, +\infty)$

Logarithm Laws

For $a > 0, a \neq 1$; $M > 0, N > 0$:

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

$\log_a M^n = n\log_a M$

4. Change of Base Formula

$\log_a b = \frac{\log_c b}{\log_c a}$

5. Reciprocal Relationship

$\log_a b \cdot \log_b a = 1$

6. Chain Rule

$\log_a b \cdot \log_b c = \log_a c$

Special Logarithms

Common Logarithm

Base 10, denoted $\lg x$: $\lg x = \log_{10} x$

Natural Logarithm

Base $e$ ($e \approx 2.71828$), denoted $\ln x$: $\ln x = \log_e x$

CSCA Practice Problems

[Example 1] Basic (Difficulty ★★☆☆☆)

Calculate: $\log_2 8 + \log_3 27$

Solution: $\log_2 8 = \log_2 2^3 = 3$ $\log_3 27 = \log_3 3^3 = 3$ $\text{Answer: } 6$

Common Misconceptions

❌ Misconception 1: Wrong logarithm law

Wrong: $\log_a (M + N) = \log_a M + \log_a N$

Correct: $\log_a (MN) = \log_a M + \log_a N$ (product rule)

❌ Misconception 2: Forgetting argument must be positive

Wrong: $\log_2 (-4)$ is defined

Correct: Argument must be $> 0$, so $\log_2 (-4)$ is undefined

Study Tips

1. ✅ Understand definition: Logarithm is inverse of exponentiation
2. ✅ Memorize laws: Product, quotient, power, change of base
3. ✅ Check domain: Argument $> 0$, base $> 0$ and $\neq 1$
4. ✅ Practice: Master calculation techniques

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💡 Exam Tip: Logarithmic functions are mandatory in CSCA! Must master logarithm laws. Accounts for about 20% of function problems.