Grade 11
  1. Algebra  1.1. Sequences and Series  1.1.1. Arithmetic Sequence  1.1.2. Arithmetic Series  1.1.3. Geometric Sequence  1.1.4. Geometric Series  1.1.5. Convergence and Divergence  1.1.6. Sum to Infinity  1.1.7. Sigma Notation  1.2. Complex Numbers  1.2.1. Imaginary and Complex Numbers  1.2.2. Operations with Complex Numbers  1.2.3. Polar and Cartesian Forms  1.2.4. Complex Conjugates  1.2.5. Modulus and Argument  1.2.6. De Moivre's Theorem  1.2.7. Powers and Roots of Complex Numbers  1.3. Binomial Theorem  1.3.1. Expansion of Binomials  1.3.2. General Term  1.3.3. Binomial Coefficients  1.3.4. Applications of Binomial Theorem  1.3.5. Pascal's Triangle
  2. Functions and Graphs  2.1. Types of Functions  2.1.1. Polynomial Functions  2.1.2. Rational Functions  2.1.3. Exponential Functions  2.1.4. Logarithmic Functions  2.1.5. Trigonometric Functions  2.1.6. Piecewise Functions  2.2. Transformations of Functions  2.2.1. Translations  2.2.2. Reflections  2.2.3. Dilations and Stretches  2.2.4. Rotations  2.3. Inverse Functions  2.3.1. Finding Inverses  2.3.2. Graphs of Inverse Functions  2.3.3. Properties of Inverse Functions  2.3.4. Restrictions for Inverses
  3. Trigonometry  3.1. Trigonometric Ratios and Identities  3.1.1. Basic Trigonometric Ratios  3.1.2. Pythagorean Identities  3.1.3. Sum and Difference Formulas  3.1.4. Double Angle Formulas  3.1.5. Half Angle Formulas  3.1.6. Product-to-Sum and Sum-to-Product Formulas  3.2. Trigonometric Equations  3.2.1. Solving Trigonometric Equations  3.2.2. General Solutions in Trigonometric Equations  3.3. Graphs of Trigonometric Functions  3.3.1. Sine Graph  3.3.2. Cosine Graph  3.3.3. Tangent Graph  3.3.4. Amplitude and Period Adjustments  3.3.5. Phase Shifts in Graphs of Trigonometric Functions  3.4. Applications of Trigonometry  3.4.1. Laws of Sines and Cosines  3.4.2. Area of Triangles  3.4.3. Solving Triangles  3.4.4. Bearings and Navigation
  4. Calculus  4.1. Limits and Continuity  4.1.1. Concept of a Limit  4.1.2. One-Sided Limits  4.1.3. Limits at Infinity  4.1.4. Continuity of a Function  4.1.5. Types of Discontinuities  4.2. Differentiation  4.2.1. Definition of Derivative  4.2.2. Rules of Differentiation  4.2.3. Higher Order Derivatives  4.2.4. Chain Rule  4.2.5. Product Rule  4.2.6. Quotient Rule  4.3. Applications of Differentiation  4.3.1. Rate of Change  4.3.2. Tangents and Normals  4.3.3. Maxima and Minima  4.3.4. Increasing and Decreasing Functions  4.3.5. Optimization Problems  4.3.6. Curve Sketching  4.3.7. Related Rates  4.4. Integration  4.4.1. Antiderivatives  4.4.2. Indefinite Integrals  4.4.3. Definite Integrals  4.4.4. Fundamental Theorem of Calculus  4.4.5. Techniques of Integration  4.4.6. Area Under a Curve  4.5. Applications of Integration  4.5.1. Area Between Curves  4.5.2. Volumes of Solids of Revolution  4.5.3. Average Value of a Function
  5. Vectors and Matrices  5.1. Vectors  5.1.1. Representation of Vectors  5.1.2. Vector Operations  5.1.3. Dot Product  5.1.4. Cross Product  5.1.5. Applications in Geometry  5.1.6. Projection of Vectors  5.2. Matrices  5.2.1. Types of Matrices  5.2.2. Matrix Operations  5.2.3. Determinants  5.2.4. Inverse of a Matrix  5.2.5. Solving Systems of Linear Equations  5.2.6. Cramer's Rule
  6. Probability and Statistics  6.1. Probability  6.1.1. Basic Probability Concepts  6.1.2. Conditional Probability  6.1.3. Independent and Dependent Events  6.1.4. Bayes' Theorem  6.1.5. Combinatorial Probability  6.2. Random Variables  6.2.1. Discrete Random Variables  6.2.2. Continuous Random Variables  6.2.3. Probability Distributions  6.3. Probability Distributions  6.3.1. Binomial Distribution  6.3.2. Normal Distribution  6.3.3. Poisson Distribution  6.3.4. Uniform Distribution  6.4. Statistics  6.4.1. Measures of Central Tendency  6.4.2. Measures of Dispersion  6.4.3. Data Collection and Representation  6.4.4. Correlation and Regression  6.4.5. Sampling Techniques
  7. Coordinate Geometry  7.1. Straight Lines  7.1.1. Slope and Intercept Form  7.1.2. Distance Formula  7.1.3. Midpoint Formula  7.1.4. Angle Between Lines  7.1.5. Equation of Lines  7.2. Conic Sections  7.2.1. Circle  7.2.2. Parabola  7.2.3. Ellipse  7.2.4. Hyperbola  7.2.5. Properties of Conics  7.2.6. Parametric Equations of Conics  7.2.7. Polar Coordinates of Conics
  8. Mathematical Reasoning  8.1. Logic  8.1.1. Statements and Logical Connectives  8.1.2. Truth Tables  8.1.3. Logical Equivalence  8.1.4. Quantifiers  8.1.5. Proof Techniques  8.2. Proofs  8.2.1. Direct Proof  8.2.2. Indirect Proof  8.2.3. Proof by Contradiction  8.2.4. Proof by Mathematical Induction

Grade 11 → Functions and Graphs → Types of Functions ↓


Logarithmic Functions


Logarithmic functions are a fundamental part of mathematics, especially in contexts where exponential growth or decay is involved. The logarithmic function is the inverse of the exponential function. In simple terms, if you know the value of the exponent and want to find the root base, you can use the logarithmic function to find the root base. Let's use logarithms. Let's try to understand what logarithmic functions actually are, how they are drawn, and explore their applications through various examples and visual aids.

Understanding logarithmic functions

The logarithmic function is expressed as:

y = log b (x)

In this expression:

  • b is the base of the logarithm.
  • x is the argument of the logarithm.
  • y is the exponent to which x is obtained by raising the base b.

It is important to remember the relationship between logarithms and exponents. If:

 b y = x

The equivalent logarithmic form is:

y = log b (x)

Let us look at this graphically with an example.

X Y y = log 2 (x)

Common ground

Different bases are used in logarithmic functions depending on the context. The most common bases are:

  • Base 10: This is known as the common logarithm, represented as log(x), or sometimes written as log 10 (x).
  • Base e: Known as the natural logarithm, represented as ln(x). Here, e is Euler's number, which is approximately equal to 2.71828.
  • Binary logarithm: where the base b is 2, represented as log 2 (x).

Properties of logarithms

Logarithms have several important properties that make them very useful for calculations:

1. Product rule

The logarithm of a product is the sum of the logarithms of the following factors:

log b (xy) = log b (x) + log b (y)

2. Quotient rule

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

log b (x/y) = log b (x) - log b (y)

3. Power rule

The logarithm of a power is the exponent multiplied by the logarithm of the base:

log b (x n ) = n*log b (x)

4. Base change formula

Logarithms can be converted from one base to another using the base change formula. For example, to convert log b (x) to base 10:

log b (x) = log(x) / log(b)

Graphs of logarithmic functions

The graph of a logarithmic function is a curve that grows from left to right. Unlike exponential functions that increase or decrease rapidly, logarithmic functions grow continuously.

A typical graph of y = log 2 (x) is shown below:

X Y y = log 2 (x)

Key points to note about the graph:

  • The graph passes through the point (1, 0). This is because any logarithm of 1 is zero, since b 0 = 1.
  • The graph has a vertical asymptote at x = 0; it approaches the y-axis but never touches it.
  • This function is undefined for x ≤ 0, because you cannot have the logarithm of zero or a negative number.

Examples and applications

Example 1: Calculating the logarithm

Calculate log 10 (1000).

We know that 10 3 = 1000, so based on this, log 10 (1000) = 3.

Example 2: Using logarithm properties

Simplify log 2 (32) - log 2 (4)

Using the quotient rule:

log 2 (32/4) = log 2 (8)

Since 2 3 = 8, log 2 (8) = 3.

Applications in real life

Logarithmic functions are used in many real-life applications, such as:

  • Earthquake measurement: The Richter scale for earthquakes is logarithmic. A magnitude 7.0 earthquake is ten times more powerful than a magnitude 6.0 earthquake.
  • pH level: The pH scale in chemistry is a logarithmic measure of the hydrogen ion concentration in a solution. A pH of 3 is ten times more acidic than a pH of 4.
  • Sound intensity: In acoustics, the decibel (dB) measures sound intensity logarithmically.

Conclusion

Logarithmic functions are an essential mathematical tool for analyzing phenomena that show exponential growth or decay. Through the properties and behaviors of logarithms, we can better understand complex systems and datasets. By exploring different examples and by graphing these functions, we gain a more intuitive understanding of how logarithms work. Whether it's calculating the intensity of sound or navigating the chemical acidity scale, logarithmic functions are vital to theoretical mathematics and practical application. Bridges the gap in between.


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Logarithmic Functions

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