Graphing Exponential and Logarithmic Functions

This page provides visual representations and key characteristics of exponential and logarithmic functions.

Graphs of exponential functions y = a^x are shown for cases where a > 1 and 0 < a < 1. The shape and behavior of these graphs are explained, including the y-intercept at (0,1).

Logarithmic function graphs y = log_a x are presented for bases a > 1 and 0 < a < 1. The x-intercept at (1,0) is highlighted for both cases.

Example: The graph of y = 2^x is shown as an increasing exponential function, while y = log_2 x is its inverse logarithmic function.

The relationship between exponential and logarithmic forms is summarized, emphasizing that logarithms are the inverse function of exponentiation.

Vocabulary: The logarithmic form log_a N = x is equivalent to the exponential form N = a^x, where a > 0, a ≠ 1, and N > 0.

This visual approach helps students understand the behavior and relationship between exponential and logarithmic functions.

Converting Between Logarithmic and Exponential Forms

This page focuses on converting equations from logarithmic form to exponential form, a crucial skill in working with logarithms.

The general rule for conversion is presented: log_a N = x is equivalent to N = a^x, where a > 0, a ≠ 1, and N > 0.

Multiple examples are provided to illustrate this conversion process:

1. log_5 25 = 2 converts to 25 = 5²
2. log_9 3 = 1/2 converts to 3 = 9^(1/2)
3. log_10 100 = 2 converts to 100 = 10²

Example: log_7 1 = 0 converts to 1 = 7⁰, demonstrating a key property of logarithms.

The page also includes more complex examples involving variables:

Example: log_3 $2x$ = log_3 12 converts to 2x = 12, which can be solved to find x = 6.

These exercises help students practice and reinforce their understanding of the relationship between logarithmic and exponential forms.

Highlight: Mastering the conversion between logarithmic and exponential forms is essential for solving a wide range of logarithmic equations.

Laws of Logarithms

This page covers the fundamental laws of logarithms, which are crucial for manipulating and simplifying logarithmic expressions.

The laws presented include:

1. Power Rule: log_a M^n = n log_a M
2. Product Rule: log_a $MN$ = log_a M + log_a N
3. Quotient Rule: log_a $M/N$ = log_a M - log_a N

Definition: The Power Rule allows us to move an exponent from inside a logarithm to outside as a coefficient.

Multiple examples are provided to demonstrate the application of these laws:

Example: log_2 8 = log_2 2³ = 3 log_2 2 = 3 × 1 = 3

The page also includes more complex examples combining multiple laws:

Example: log_3 3 + log_3 27 = log_3 $3 × 27$ = log_3 81 = 4

These examples help students understand how to apply the laws of logarithms to simplify and solve logarithmic expressions.

Highlight: Understanding and applying the laws of logarithms is essential for solving complex logarithmic equations and expressions.

Advanced Logarithmic Expressions

This page focuses on more complex logarithmic expressions and how to simplify them using the laws of logarithms.

Students are presented with exercises that require combining multiple logarithmic terms into a single logarithm:

Example: log_2 12 + log_2 3 - 2 log_2 2 = log_2 $12 × 3 / 2²$ = log_2 9 = log_2 3²

The page also includes examples with fractional and irrational bases:

Example: log_√2 4 + log_√2 5 = log_√2 $4 × 5$ = log_√2 20

More advanced problems involve combining logarithms with other operations:

Example: 3log_4 8 + log_4 6 - 3 = log_4 8³ + log_4 6 - log_4 4³ = log_4 (512 × 6 / 64)

These exercises help students develop their skills in manipulating complex logarithmic expressions and reinforce their understanding of logarithmic laws.

Highlight: Mastering the simplification of complex logarithmic expressions is crucial for advanced problem-solving in mathematics and sciences.

Simplifying Logarithmic Expressions

This page presents advanced techniques for simplifying complex logarithmic expressions.

Students are challenged with problems that require a deep understanding of logarithmic properties and algebraic manipulation:

Example: Simplify 7^$log_x x$ = 7^1 = 7

The page includes examples that combine logarithms with other mathematical concepts:

Example: Simplify log_x $x³ - 5x² + 6x$ / $(x-3)(x-2)$ = log_x $x(x² - 5x + 6)$ / $(x-3)(x-2)$

More complex problems involve multiple bases and properties:

Example: Simplify log 35 - log 7 + log 125 + log 7 + log 3 = log $35 × 5 × 3 × 3$ / 7 = log 225

The exercises on this page help students develop advanced problem-solving skills and a deeper understanding of logarithmic properties.

Highlight: The ability to simplify complex logarithmic expressions is a valuable skill in advanced mathematics and many scientific fields.

Equations Involving Indices

This page introduces the concept of solving equations involving indices (exponents).

The content focuses on two main types of equations:

1. Equations with the same base: a^x = a^n, which simplifies to x = n
2. Equations with the same index: x^a = n^a, which simplifies to x = n

Definition: An index (or exponent) shows how many times a number is multiplied by itself.

These rules provide a foundation for solving more complex exponential equations.

Highlight: Understanding how to solve equations involving indices is crucial for working with exponential functions and logarithms.

The page sets the stage for more advanced topics in exponential and logarithmic equations, which are fundamental in many areas of mathematics and science.

Rules of Surds and Indices

This page covers fundamental rules for working with surds and indices in mathematics.

The rules of surds are presented, including key properties like √a × √b = √ab and $√a$² = a. These rules allow for simplifying and manipulating expressions containing square roots.

The laws of indices are also outlined, covering properties such as a⁰ = 1 and $a^m$^n = a^(mn). These laws are essential for working with exponents and simplifying exponential expressions.

Definition: Surds are irrational numbers that can be expressed as the root of a rational number.

Definition: Indices (also called exponents) indicate how many times a number is multiplied by itself.

The page concludes with brief notes on equations involving indices and rules of inequalities, providing a foundation for more advanced topics.

Highlight: Understanding the rules of surds and indices is crucial for solving more complex mathematical problems in algebra and calculus.