# 6.2 Logarithm

## Detailed Theory: Logarithms

### **1. Basic Concepts and Definitions**

#### **1.1 What is a Logarithm?**

A logarithm is the **inverse operation** to exponentiation.

If $$b^x = y$$, then $$\log\_b y = x$$

**Reading:** "log base b of y equals x"

**Example:** Since $$2^3 = 8$$, then $$\log\_2 8 = 3$$

#### **1.2 Components of a Logarithm**

In $$\log\_b a = c$$:

* **b** is the **base** (must be positive and not equal to 1)
* **a** is the **argument** (must be positive)
* **c** is the **value** or **exponent**

#### **1.3 Why Use Logarithms?**

1. **Simplify calculations:** Convert multiplication to addition
2. **Solve exponential equations**
3. **Model exponential growth/decay**
4. **Scale large ranges** (Richter scale, pH, decibels)

#### **1.4 Exponential-Logarithmic Relationship**

Logarithm and exponentiation are inverse operations:

$$
b^{\log\_b x} = x \quad \text{and} \quad \log\_b(b^x) = x
$$

**Important:** These only work when $$x > 0$$

***

### **2. Types of Logarithms**

#### **2.1 Common Logarithms (Base 10)**

* Base = 10
* Notation: $$\log x$$ (base 10 implied)
* Also called Briggsian logarithms
* Used in scientific calculations

**Example:** $$\log 100 = 2$$ because $$10^2 = 100$$

#### **2.2 Natural Logarithms (Base e)**

* Base = $$e \approx 2.71828$$
* Notation: $$\ln x$$
* Also called Napierian logarithms
* Used in calculus and higher mathematics

**Example:** $$\ln e = 1$$ because $$e^1 = e$$

#### **2.3 Binary Logarithms (Base 2)**

* Base = 2
* Notation: $$\log\_2 x$$ or $$\text{lb } x$$
* Used in computer science and information theory

**Example:** $$\log\_2 8 = 3$$ because $$2^3 = 8$$

#### **2.4 Relationship Between Different Bases**

For any positive numbers $$a, b, x$$ (with $$a, b \neq 1$$):

$$
\log\_a x = \frac{\log\_b x}{\log\_b a}
$$

This is called the **change of base formula**.

**Special case:** $$\log\_a x = \frac{\ln x}{\ln a} = \frac{\log x}{\log a}$$

***

### **3. Fundamental Properties and Laws**

#### **3.1 Basic Properties**

**a) Logarithm of 1**

For any base $$b > 0$$, $$b \neq 1$$:

$$
\log\_b 1 = 0
$$

**Reason:** $$b^0 = 1$$

**b) Logarithm of Base**

$$
\log\_b b = 1
$$

**Reason:** $$b^1 = b$$

**c) Logarithm of Power of Base**

$$
\log\_b (b^k) = k
$$

**Example:** $$\log\_2 (2^5) = 5$$

**d) Power of Logarithm**

$$
b^{\log\_b x} = x
$$

**Example:** $$10^{\log 100} = 100$$

#### **3.2 Three Fundamental Laws**

**a) Product Law**

$$
\log\_b (mn) = \log\_b m + \log\_b n
$$

**Verification:** Let $$\log\_b m = x$$ and $$\log\_b n = y$$

Then $$m = b^x$$ and $$n = b^y$$

So $$mn = b^x \cdot b^y = b^{x+y}$$

Thus $$\log\_b (mn) = x + y = \log\_b m + \log\_b n$$

**Example:** $$\log\_2 (4 \times 8) = \log\_2 4 + \log\_2 8 = 2 + 3 = 5$$

**b) Quotient Law**

$$
\log\_b \left(\frac{m}{n}\right) = \log\_b m - \log\_b n
$$

**Verification:** Let $$\log\_b m = x$$ and $$\log\_b n = y$$

Then $$m = b^x$$ and $$n = b^y$$

So $$\frac{m}{n} = \frac{b^x}{b^y} = b^{x-y}$$

Thus $$\log\_b \left(\frac{m}{n}\right) = x - y = \log\_b m - \log\_b n$$

**Example:** $$\log\_3 \left(\frac{27}{9}\right) = \log\_3 27 - \log\_3 9 = 3 - 2 = 1$$

**c) Power Law**

$$
\log\_b (m^n) = n \log\_b m
$$

**Verification:** Let $$\log\_b m = x$$

Then $$m = b^x$$

So $$m^n = (b^x)^n = b^{nx}$$

Thus $$\log\_b (m^n) = nx = n \log\_b m$$

**Example:** $$\log\_2 (8^3) = 3 \log\_2 8 = 3 \times 3 = 9$$

#### **3.3 Additional Important Properties**

**a) Change of Base**

$$
\log\_a b = \frac{1}{\log\_b a}
$$

**Proof:** Using change of base formula:

$$
\log\_a b = \frac{\log\_b b}{\log\_b a} = \frac{1}{\log\_b a}
$$

**b) Chain Rule**

$$
\log\_a b \times \log\_b c = \log\_a c
$$

**Proof:**

$$
\log\_a b \times \log\_b c = \frac{\log b}{\log a} \times \frac{\log c}{\log b} = \frac{\log c}{\log a} = \log\_a c
$$

**c) Root Property**

$$
\log\_b \sqrt\[n]{m} = \frac{1}{n} \log\_b m
$$

**Reason:** $$\sqrt\[n]{m} = m^{1/n}$$, so apply power law

**Example:** $$\log \sqrt{100} = \frac{1}{2} \log 100 = \frac{1}{2} \times 2 = 1$$

***

### **4. Domain and Range**

#### **4.1 Domain Restrictions**

For $$\log\_b x$$ to be defined:

1. **Base b:** $$b > 0$$ and $$b \neq 1$$
   * If $$b = 1$$: $$1^x$$ is always 1, so inverse not unique
   * If $$b = 0$$: $$0^x$$ is 0 for $$x > 0$$, undefined for $$x \leq 0$$
   * If $$b < 0$$: Results become complex
2. **Argument x:** $$x > 0$$
   * You cannot take log of zero or negative numbers in real numbers

#### **4.2 Range**

For $$\log\_b x$$:

* If $$b > 1$$: Range is all real numbers ($$-\infty, \infty$$)
* If $$0 < b < 1$$: Range is all real numbers ($$-\infty, \infty$$)

#### **4.3 Example: Find Domain of** $$\log\_2 (x-3)$$

Argument must be positive: $$x - 3 > 0$$

So $$x > 3$$

Domain: $$(3, \infty)$$

***

### **5. Graphs of Logarithmic Functions**

#### **5.1 Basic Shape**

The graph of $$y = \log\_b x$$ has:

1. **Vertical asymptote** at $$x = 0$$ (y-axis)
2. **x-intercept** at $$x = 1$$ (since $$\log\_b 1 = 0$$)
3. **Passes through** point $$(b, 1)$$ (since $$\log\_b b = 1$$)

#### **5.2 Two Cases**

**Case 1: Base b > 1**

* Function is **increasing**
* As $$x \to 0^+$$, $$y \to -\infty$$
* As $$x \to \infty$$, $$y \to \infty$$
* **Example:** $$y = \log\_2 x$$, $$y = \ln x$$

**Case 2: Base 0 < b < 1**

* Function is **decreasing**
* As $$x \to 0^+$$, $$y \to \infty$$
* As $$x \to \infty$$, $$y \to -\infty$$
* **Example:** $$y = \log\_{0.5} x$$

#### **5.3 Transformations**

**a) Vertical Shift**

$$y = \log\_b x + c$$ shifts graph up by c units

**b) Horizontal Shift**

$$y = \log\_b (x - h)$$ shifts graph right by h units

**c) Vertical Stretch/Compression**

$$y = a \log\_b x$$ multiplies y-values by a

**d) Horizontal Stretch/Compression**

$$y = \log\_b (kx)$$ affects x-values

#### **5.4 Example: Graph** $$y = 2\log\_3 (x-1) + 1$$

1. Start with basic $$y = \log\_3 x$$
2. Shift right 1 unit: $$y = \log\_3 (x-1)$$
3. Vertical stretch by 2: $$y = 2\log\_3 (x-1)$$
4. Shift up 1 unit: $$y = 2\log\_3 (x-1) + 1$$

Domain: $$x-1 > 0 \Rightarrow x > 1$$

Vertical asymptote: $$x = 1$$

x-intercept: Set $$y=0$$: $$2\log\_3 (x-1) + 1 = 0$$

$$\log\_3 (x-1) = -\frac{1}{2}$$

$$x-1 = 3^{-1/2} = \frac{1}{\sqrt{3}}$$

$$x = 1 + \frac{1}{\sqrt{3}}$$

***

### **6. Solving Logarithmic Equations**

#### **6.1 Basic Methods**

**a) Using Definition**

Convert to exponential form.

**Example:** Solve $$\log\_2 x = 3$$

By definition: $$x = 2^3 = 8$$

**b) Using Properties**

Combine logs using product/quotient/power laws.

**Example:** Solve $$\log x + \log (x-3) = 1$$

Using product law: $$\log \[x(x-3)] = 1$$

So $$x(x-3) = 10^1 = 10$$

$$x^2 - 3x - 10 = 0$$

$$(x-5)(x+2) = 0$$

$$x = 5$$ or $$x = -2$$

But domain requires $$x > 0$$ and $$x-3 > 0$$, so $$x > 3$$

Thus $$x = 5$$ is only solution.

**c) Using Change of Base**

Make all logs have same base.

**Example:** Solve $$\log\_2 x = \log\_4 9$$

Change base: $$\log\_4 9 = \frac{\log\_2 9}{\log\_2 4} = \frac{\log\_2 9}{2}$$

So equation becomes: $$\log\_2 x = \frac{\log\_2 9}{2}$$

Multiply by 2: $$2\log\_2 x = \log\_2 9$$

$$\log\_2 (x^2) = \log\_2 9$$

Thus $$x^2 = 9$$, so $$x = 3$$ or $$x = -3$$

Since $$x > 0$$, $$x = 3$$

#### **6.2 Checking for Extraneous Solutions**

**IMPORTANT:** Always check solutions in original equation!

Reasons for extraneous solutions:

1. Argument of log becomes non-positive
2. Base becomes invalid

**Example:** Solve $$\log\_2 (x-1) + \log\_2 (x+1) = 3$$

Using product law: $$\log\_2 \[(x-1)(x+1)] = 3$$

$$(x-1)(x+1) = 2^3 = 8$$

$$x^2 - 1 = 8$$

$$x^2 = 9$$

$$x = 3$$ or $$x = -3$$

Check domain: $$x-1 > 0$$ and $$x+1 > 0$$

For $$x = 3$$: $$3-1=2>0$$, $$3+1=4>0$$ ✓

For $$x = -3$$: $$-3-1=-4<0$$ ✗

So only solution is $$x = 3$$

***

### **7. Solving Exponential Equations Using Logarithms**

#### **7.1 Basic Method**

To solve $$a^x = b$$:

1. Take log of both sides: $$\log(a^x) = \log b$$
2. Use power law: $$x \log a = \log b$$
3. Solve for x: $$x = \frac{\log b}{\log a}$$

#### **7.2 Examples**

**Example 1: Solve** $$2^x = 5$$

Take log: $$\log(2^x) = \log 5$$

$$x \log 2 = \log 5$$

$$x = \frac{\log 5}{\log 2} \approx \frac{0.6990}{0.3010} \approx 2.322$$

**Example 2: Solve** $$3^{2x-1} = 7^{x+2}$$

Take natural log: $$\ln(3^{2x-1}) = \ln(7^{x+2})$$

$$(2x-1)\ln 3 = (x+2)\ln 7$$

Expand: $$2x\ln 3 - \ln 3 = x\ln 7 + 2\ln 7$$

Group x terms: $$2x\ln 3 - x\ln 7 = 2\ln 7 + \ln 3$$

Factor: $$x(2\ln 3 - \ln 7) = 2\ln 7 + \ln 3$$

$$x = \frac{2\ln 7 + \ln 3}{2\ln 3 - \ln 7}$$

**Example 3: Solve** $$e^{2x} - 5e^x + 6 = 0$$

Let $$y = e^x$$, then equation becomes:

$$y^2 - 5y + 6 = 0$$

$$(y-2)(y-3) = 0$$

So $$y = 2$$ or $$y = 3$$

Thus $$e^x = 2$$ or $$e^x = 3$$

$$x = \ln 2$$ or $$x = \ln 3$$

***

### **8. Logarithmic Inequalities**

#### **8.1 Important Consideration**

When solving $$\log\_b f(x) > \log\_b g(x)$$:

**Case 1: If b > 1** (function increasing) Then $$f(x) > g(x)$$ **AND** $$f(x) > 0$$, $$g(x) > 0$$

**Case 2: If 0 < b < 1** (function decreasing) Then $$f(x) < g(x)$$ **AND** $$f(x) > 0$$, $$g(x) > 0$$

#### **8.2 Examples**

**Example 1: Solve** $$\log\_2 (x-1) < 3$$

Since base 2 > 1, inequality direction preserved:

First, domain: $$x-1 > 0 \Rightarrow x > 1$$

Inequality: $$\log\_2 (x-1) < 3$$

Convert: $$x-1 < 2^3 = 8$$

So $$x < 9$$

Combining with domain: $$1 < x < 9$$

**Example 2: Solve** $$\log\_{0.5} (2x-1) > -1$$

Base 0.5 is between 0 and 1, so inequality reverses:

First, domain: $$2x-1 > 0 \Rightarrow x > \frac{1}{2}$$

Inequality: $$\log\_{0.5} (2x-1) > -1$$

Since base < 1, reverse: $$2x-1 < (0.5)^{-1} = 2$$

So $$2x-1 < 2 \Rightarrow 2x < 3 \Rightarrow x < \frac{3}{2}$$

Combining with domain: $$\frac{1}{2} < x < \frac{3}{2}$$

***

### **9. Applications of Logarithms**

#### **9.1 Scientific Applications**

**a) Richter Scale (Earthquakes)**

Magnitude $$M = \log\_{10} \left(\frac{A}{A\_0}\right)$$

where A is amplitude, $$A\_0$$ is reference amplitude.

**Example:** Earthquake 1000 times stronger than reference:

$$M = \log\_{10} 1000 = 3$$

**b) pH Scale (Acidity)**

$$\text{pH} = -\log\_{10} \[\text{H}^+]$$

where $$\[\text{H}^+]$$ is hydrogen ion concentration.

**Example:** Solution with $$\[\text{H}^+] = 10^{-5}$$ M:

$$\text{pH} = -\log\_{10} (10^{-5}) = -(-5) = 5$$

**c) Decibel Scale (Sound)**

$$\text{dB} = 10 \log\_{10} \left(\frac{I}{I\_0}\right)$$

where I is intensity, $$I\_0$$ is reference intensity.

**Example:** Sound 100 times more intense than reference:

$$\text{dB} = 10 \log\_{10} 100 = 10 \times 2 = 20 \text{ dB}$$

#### **9.2 Mathematical Applications**

**a) Solving Compound Interest**

For principal P, rate r, time t, compounded n times per year:

$$A = P\left(1 + \frac{r}{n}\right)^{nt}$$

To find time to reach amount A:

$$t = \frac{\log(A/P)}{n \log(1 + r/n)}$$

**b) Half-life Problems**

For exponential decay: $$N = N\_0 e^{-kt}$$

Half-life $$t\_{1/2} = \frac{\ln 2}{k}$$

**c) Logarithmic Scales**

* Used when data spans many orders of magnitude
* Examples: star magnitudes, financial charts (log scale)

***

### **10. Natural Logarithms and Calculus**

#### **10.1 Definition of e**

The number e is defined as:

$$
e = \lim\_{n \to \infty} \left(1 + \frac{1}{n}\right)^n \approx 2.71828
$$

#### **10.2 Derivative of ln x**

$$
\frac{d}{dx} (\ln x) = \frac{1}{x}, \quad x > 0
$$

#### **10.3 Integral of 1/x**

$$
\int \frac{1}{x} , dx = \ln |x| + C
$$

#### **10.4 Derivatives of General Logarithms**

Using change of base: $$\log\_b x = \frac{\ln x}{\ln b}$$

So $$\frac{d}{dx} (\log\_b x) = \frac{1}{x \ln b}$$

#### **10.5 Logarithmic Differentiation**

Technique for differentiating complicated functions:

**Steps:**

1. Take ln of both sides: $$\ln y = \ln f(x)$$
2. Differentiate implicitly: $$\frac{y'}{y} = \frac{d}{dx}\[\ln f(x)]$$
3. Solve for y'

**Example:** Find derivative of $$y = x^x$$

Take ln: $$\ln y = \ln(x^x) = x \ln x$$

Differentiate: $$\frac{y'}{y} = 1 \cdot \ln x + x \cdot \frac{1}{x} = \ln x + 1$$

So $$y' = y(\ln x + 1) = x^x (\ln x + 1)$$

***

### **11. Logarithmic Series and Limits**

#### **11.1 Series Expansion for ln(1+x)**

For $$-1 < x \leq 1$$:

$$
\ln(1+x) = x - \frac{x^2}{2} + \frac{x^3}{3} - \frac{x^4}{4} + \cdots = \sum\_{n=1}^{\infty} (-1)^{n-1} \frac{x^n}{n}
$$

#### **11.2 Important Limits**

1. $$\lim\_{x \to 0} \frac{\ln(1+x)}{x} = 1$$
2. $$\lim\_{x \to \infty} \frac{\ln x}{x} = 0$$
3. $$\lim\_{x \to 0^+} x \ln x = 0$$
4. $$\lim\_{x \to \infty} \left(1 + \frac{1}{x}\right)^x = e$$

#### **11.3 Example: Evaluate** $$\lim\_{x \to 0} \frac{\ln(\cos x)}{x^2}$$

Using series: $$\cos x = 1 - \frac{x^2}{2} + \cdots$$

So $$\ln(\cos x) = \ln\left(1 - \frac{x^2}{2} + \cdots\right) \approx -\frac{x^2}{2}$$ for small x

Thus $$\lim\_{x \to 0} \frac{\ln(\cos x)}{x^2} = \lim\_{x \to 0} \frac{-x^2/2}{x^2} = -\frac{1}{2}$$

***

### **12. Common Mistakes and Pitfalls**

#### **12.1 Most Common Errors**

1. $$\log\_b (m+n) \neq \log\_b m + \log\_b n$$
   * Correct: $$\log\_b (mn) = \log\_b m + \log\_b n$$
2. $$\log\_b (m-n) \neq \log\_b m - \log\_b n$$
   * Correct: $$\log\_b \left(\frac{m}{n}\right) = \log\_b m - \log\_b n$$
3. $$(\log\_b m)^n \neq \log\_b (m^n)$$
   * Correct: $$\log\_b (m^n) = n \log\_b m$$
4. $$\log\_b m \times \log\_b n \neq \log\_b (mn)$$
   * Correct: $$\log\_b m + \log\_b n = \log\_b (mn)$$
5. **Forgetting domain restrictions**
   * Argument must be positive
   * Base must be positive and not 1

#### **12.2 Example of Common Error**

**Wrong:** $$\log\_2 (8 + 4) = \log\_2 8 + \log\_2 4 = 3 + 2 = 5$$

**Right:** $$\log\_2 (8 + 4) = \log\_2 12 = \log\_2 (4 \times 3) = \log\_2 4 + \log\_2 3 = 2 + \log\_2 3 \approx 3.585$$

***

### **13. Solved Examples for Practice**

#### **Example 1:** Simplify $$\frac{\log\_2 27}{\log\_2 3}$$

**Solution:**

Using change of base: $$\log\_2 27 = \log\_2 (3^3) = 3\log\_2 3$$

So $$\frac{\log\_2 27}{\log\_2 3} = \frac{3\log\_2 3}{\log\_2 3} = 3$$

#### **Example 2:** Solve $$2\log x = \log 4 + \log 16$$

**Solution:**

Right side: $$\log 4 + \log 16 = \log (4 \times 16) = \log 64$$

So $$2\log x = \log 64$$

$$\log (x^2) = \log 64$$

Thus $$x^2 = 64$$

$$x = 8$$ or $$x = -8$$

Since argument must be positive: $$x = 8$$

#### **Example 3:** Evaluate $$\log\_3 54 - \log\_3 8 + \log\_3 4$$

**Solution:**

Using properties: $$\log\_3 54 - \log\_3 8 + \log\_3 4 = \log\_3 \left(\frac{54 \times 4}{8}\right)$$

$$= \log\_3 \left(\frac{216}{8}\right) = \log\_3 27 = \log\_3 (3^3) = 3$$

#### **Example 4:** If $$\log\_{10} 2 = 0.3010$$, find $$\log\_{10} 8$$

**Solution:**

$$\log\_{10} 8 = \log\_{10} (2^3) = 3\log\_{10} 2 = 3 \times 0.3010 = 0.9030$$

#### **Example 5:** Solve $$2^{x+1} = 5^{x-1}$$

**Solution:**

Take log: $$\log(2^{x+1}) = \log(5^{x-1})$$

$$(x+1)\log 2 = (x-1)\log 5$$

Expand: $$x\log 2 + \log 2 = x\log 5 - \log 5$$

Group x terms: $$x\log 2 - x\log 5 = -\log 5 - \log 2$$

$$x(\log 2 - \log 5) = -(\log 5 + \log 2)$$

$$x = \frac{-(\log 5 + \log 2)}{\log 2 - \log 5} = \frac{\log 5 + \log 2}{\log 5 - \log 2} = \frac{\log 10}{\log(5/2)}$$

***

### **14. Important Formulas Summary**

#### **14.1 Fundamental Laws**

1. **Product:** $$\log\_b (mn) = \log\_b m + \log\_b n$$
2. **Quotient:** $$\log\_b \left(\frac{m}{n}\right) = \log\_b m - \log\_b n$$
3. **Power:** $$\log\_b (m^n) = n \log\_b m$$

#### **14.2 Special Values**

1. $$\log\_b 1 = 0$$
2. $$\log\_b b = 1$$
3. $$\log\_b (b^k) = k$$
4. $$b^{\log\_b x} = x$$

#### **14.3 Change of Base**

$$
\log\_a x = \frac{\log\_b x}{\log\_b a} = \frac{\ln x}{\ln a} = \frac{\log x}{\log a}
$$

#### **14.4 Domain and Range**

* **Domain:** $$x > 0$$
* **Base:** $$b > 0$$, $$b \neq 1$$
* **Range:** All real numbers

#### **14.5 Common Logarithms**

* $$\log 10 = 1$$
* $$\log 100 = 2$$
* $$\log 1000 = 3$$
* $$\log 0.1 = -1$$
* $$\log 0.01 = -2$$

#### **14.6 Natural Logarithms**

* $$\ln e = 1$$
* $$\ln 1 = 0$$
* $$\ln e^x = x$$
* $$e^{\ln x} = x$$

***

### **15. Exam Tips and Strategies**

#### **15.1 Problem-Solving Approach**

1. **Identify the type:** Equation, inequality, simplification, application?
2. **Check domain:** Ensure arguments are positive
3. **Use properties:** Combine/expand using laws
4. **Convert forms:** Exponential ↔ Logarithmic
5. **Verify solution:** Check in original equation

#### **15.2 Common Exam Questions**

1. **Simplify expressions** using logarithmic properties
2. **Solve equations** involving logs and exponents
3. **Prove identities** using logarithmic laws
4. **Application problems** (pH, Richter scale, compound interest)
5. **Graph transformations** of logarithmic functions

#### **15.3 Quick Checks**

1. **For** $$\log\_b x$$**:** Is $$x > 0$$? Is $$b > 0$$ and $$b \neq 1$$?
2. **When solving:** Did you consider all possible solutions?
3. **For inequalities:** Did you check if base > 1 or 0 < base < 1?
4. **For applications:** Are units consistent?

#### **15.4 Memory Aids**

* **Product law:** "Log of product = Sum of logs"
* **Quotient law:** "Log of quotient = Difference of logs"
* **Power law:** "Exponent comes down as multiplier"
* **Domain:** "Can't take log of zero or negative"

This comprehensive theory covers all aspects of logarithms with detailed explanations and examples, providing complete preparation for the entrance examination.