7.1 Definition and graph of logarithm

7.1 Definition and Graph of Logarithm

Definition of Logarithm

A logarithm is the inverse operation to exponentiation. The logarithm of a number is the exponent to which a given base must be raised to produce that number.

If $b^y = x$ , then:

$log_b(x) = y$ Where:

$b$ is the base (must be a positive number, and $\neq 1$ ),
$x$ is the number for which we want to find the logarithm (must be positive),
$y$ is the logarithm of $x$ to the base $b$ .

Common Logarithms

Base 10 (Common Logarithm): $log_{10}(x)$ , often written as $log⁡(x)\log(x)$ .
Base $e$ (Natural Logarithm): $log_e(x)$ , often written as $ln⁡(x)\ln(x)$ .

Properties of Logarithms

Product Rule: $log_b(xy) = \log_b(x) + \log_b(y)$
Quotient Rule: $log⁡b(xy)=log⁡b(x)−log⁡b(y)\log_b\left(\frac{x}{y}\right) = \log_b(x) – \log_b(y)$
Power Rule: $log⁡b(xn)=n⋅log⁡b(x)\log_b(x^n) = n \cdot \log_b(x)$
Change of Base Formula: $k≠1\log_b(x) = \frac{\log_k(x)}{\log_k(b)} \quad \text{for any positive } k \neq 1$

Graph of Logarithm

Key Characteristics

The graph of $y=log⁡b(x)y = \log_b(x)$ $y = log_{b} (x)$ has the following characteristics:
- Domain: $x > 0$ (logarithms are only defined for positive $x$ ).
- Range: All real numbers ( $−∞,+∞-\infty, +\infty$ ).
- Intercept: The graph crosses the x-axis at $x = 1$ (since $log_b(1) = 0$ ).
- Asymptote: There is a vertical asymptote at $x = 0$ (the graph approaches the y-axis but never touches it).
- Increasing Function: The logarithmic function is increasing for all $x > 0$ if $b > 1$ , and decreasing if $0 < b < 1$ .

Example Graph

Base 10 ( $y=log⁡10(x)y = \log_{10}(x)$ $y = log_{10} (x)$ ):
- The graph starts from negative infinity as $x$ approaches 0, crosses the x-axis at (1, 0), and continues to increase without bound.
Base $e$ ( $y=ln⁡(x)y = \ln(x)$ $y = ln (x)$ ):
- Similar to the base 10 logarithm but steeper since $\approx 2.718$ .

Graphical Representation

Here’s a simple representation of the graph:

y
↑
| *
| *
| *
| *
| *
|____________________→ x
1

The curve approaches the y-axis (vertical asymptote) and increases to the right.

7.2 Properties of logarithm

Logarithms have several important properties that facilitate their manipulation and application in various mathematical contexts. Here are the key properties:

1. Product Property

The logarithm of a product is the sum of the logarithms of the factors.

$log_b(xy) = \log_b(x) + \log_b(y)$

2. Quotient Property

The logarithm of a quotient is the difference of the logarithms.

$log⁡b(xy)=log⁡b(x)−log⁡b(y)\log_b\left(\frac{x}{y}\right) = \log_b(x) – \log_b(y)$

3. Power Property

The logarithm of a number raised to a power is the exponent times the logarithm of the base.

$log⁡b(xn)=n⋅log⁡b(x)\log_b(x^n) = n \cdot \log_b(x)$

4. Change of Base Formula

This property allows you to change the base of the logarithm to another base.

$k≠1\log_b(x) = \frac{\log_k(x)}{\log_k(b)} \quad \text{for any positive } k \neq 1$

5. Logarithm of 1

The logarithm of 1 in any base is always zero because any number raised to the power of 0 is 1.

$log_b(1) = 0$

6. Logarithm of the Base

The logarithm of a base to itself is always 1.

$log_b(b) = 1$

7. Logarithm of a Reciprocal

The logarithm of a reciprocal can be expressed as the negative of the logarithm.

$log⁡b(1x)=−log⁡b(x)\log_b\left(\frac{1}{x}\right) = -\log_b(x)$

8. Inverse Relationship

Logarithms and exponentials are inverse functions. If $y = \log_b(x)$ , then it follows that:

$b^y = x$

9. Special Cases

Natural Logarithm: The natural logarithm ( $ln⁡(x)\ln(x)$ ) uses base $e$ : $ln(x) = \log_e(x)$
Common Logarithm: The common logarithm ( $log⁡(x)\log(x)$ ) uses base 10: $log(x) = \log_{10}(x)$

Applications of Logarithm Properties

These properties are instrumental in solving equations, simplifying expressions, and in calculus, particularly in integration and differentiation involving logarithmic functions. They also play a crucial role in fields such as finance, biology, and computer science, especially in algorithms involving exponential growth or decay.

7.3 Change of base

The change of base formula is a useful tool in logarithms that allows you to convert a logarithm from one base to another. This can be particularly helpful when using calculators, which often only have specific bases (usually base 10 or base $e$ ).

Change of Base Formula

For any positive numbers $a$ , $b$ , and $x$ (where $\neq 1$ and $\neq 1$ ), the change of base formula is given by:

$log⁡b(x)=log⁡a(x)log⁡a(b)\log_b(x) = \frac{\log_a(x)}{\log_a(b)}$

This means that you can express the logarithm of $x$ to the base $b$ using logarithms of another base $a$ .

Steps for Changing the Base

Choose a New Base: Decide which base you want to convert to, such as base 10 or base $e$ .
Apply the Formula: Substitute $a$ and $b$ in the formula.
Calculate: Use a calculator to find the logarithms of $x$ and $b$ in the chosen base.

Example

Let’s say you want to calculate $log_2(8)$ using base 10:

Choose Base: We will use base 10 ( $a = 10$ ).
Apply the Formula:
$log⁡2(8)=log⁡10(8)log⁡10(2)\log_2(8) = \frac{\log_{10}(8)}{\log_{10}(2)}$
Calculate (using a calculator):
- $log⁡10(8)≈0.903\log_{10}(8) \approx 0.903$
- $log⁡10(2)≈0.301\log_{10}(2) \approx 0.301$
Thus,

$log⁡2(8)=0.9030.301≈3\log_2(8) = \frac{0.903}{0.301} \approx 3$

Special Cases

Natural Logarithm: If you want to change to the natural logarithm, you can express it as:
$log⁡b(x)=ln⁡(x)ln⁡(b)\log_b(x) = \frac{\ln(x)}{\ln(b)}$
Common Logarithm: For base 10:
$log⁡b(x)=log⁡(x)log⁡(b)\log_b(x) = \frac{\log(x)}{\log(b)}$

Applications

The change of base formula is widely used in:

Calculating Logarithms: When the desired base is not available on a calculator.
Solving Logarithmic Equations: It simplifies the manipulation of logarithmic expressions.
Mathematical Proofs: In higher mathematics, it aids in proving properties of logarithms and exponential functions.

7.4 Computation with logarithm

Computing with logarithms involves using their properties to simplify expressions, solve equations, and perform calculations efficiently. Here’s a guide to various computations involving logarithms.

1. Basic Logarithmic Calculations

To compute logarithms, you can use the following approaches:

Direct Calculation: Using a calculator that supports logarithms.
Change of Base: If the base of the logarithm is not available, use the change of base formula: $log⁡b(x)=log⁡k(x)log⁡k(b)\log_b(x) = \frac{\log_k(x)}{\log_k(b)}$ where $k$ can be any base, commonly 10 or $e$ .

2. Using Logarithmic Properties

Logarithmic properties can simplify complex computations:

Product Property: $log_b(xy) = \log_b(x) + \log_b(y)$
Quotient Property: $log⁡b(xy)=log⁡b(x)−log⁡b(y)\log_b\left(\frac{x}{y}\right) = \log_b(x) – \log_b(y)$
Power Property: $log⁡b(xn)=n⋅log⁡b(x)\log_b(x^n) = n \cdot \log_b(x)$

3. Example Computations

Example 1: Calculate $log_2(32)$

Recognize that $32 = 2^5$ .
Apply the power property: $log⁡2(32)=log⁡2(25)=5⋅log⁡2(2)=5⋅1=5\log_2(32) = \log_2(2^5) = 5 \cdot \log_2(2) = 5 \cdot 1 = 5$

Example 2: Calculate $log_{10}(1000) + \log_{10}(10)$

Use the product property: $log⁡10(1000)+log⁡10(10)=log⁡10(1000⋅10)=log⁡10(10000)\log_{10}(1000) + \log_{10}(10) = \log_{10}(1000 \cdot 10) = \log_{10}(10000)$
Since $10000 = 10^4$ : $log_{10}(10000) = 4$

Example 3: Solve $2 \log_3(4) – \log_3(16)$

Express $16$ as $4^2$ : $log⁡3(16)=log⁡3(42)=2⋅log⁡3(4)\log_3(16) = \log_3(4^2) = 2 \cdot \log_3(4)$
Substitute: $2 \log_3(4) – 2 \log_3(4) = 0$

4. Solving Logarithmic Equations

When solving equations involving logarithms, use properties to isolate the logarithmic term or convert to exponential form.

Example: Solve $log_2(x) + \log_2(4) = 6$

Use the product property: $log_2(4x) = 6$
Convert to exponential form: $2^6 \implies 4x = 64 \implies x = \frac{64}{4} = 16$

5. Applications of Logarithms in Calculations

Logarithms are widely used in:

Exponential Growth/Decay Models: In fields like biology, finance, and physics.
Complexity Analysis: In computer science for algorithms.
pH Calculations: In chemistry (pH = $−log⁡10[H+]-\log_{10}[\text{H}^+]$ ).
Sound Intensity: Measuring decibels (dB) involves logarithms.

Logarithms

7.1 Definition and graph of logarithm