Indices, Surds, and Logarithms Cheat Sheet

This cheat sheet covers the fundamental concepts of indices, surds, and logarithms, including their properties, simplification techniques, and applications in solving equations. Understanding these topics is crucial for advanced mathematics and various scientific fields.

Core Principles

Indices (Exponents): Rules for multiplication, division, powers of powers, zero and negative exponents, and fractional exponents.
Surds (Radicals): Understanding irrational numbers expressed with root signs, simplifying surds, and performing operations like addition, subtraction, multiplication, and division.
Logarithms: The relationship between exponential and logarithmic forms, evaluating logarithms, and applying logarithm laws to simplify expressions and solve equations.
Rationalizing Denominators: Techniques to remove surds from the denominator of a fraction, often using the difference of squares.
Solving Equations: Applying the properties of indices, surds, and logarithms to solve various types of equations.

Action Steps

When simplifying indices, apply the relevant laws (product, quotient, power of a power, etc.).
To simplify surds, find the largest perfect square factor under the root sign.
When adding or subtracting surds, combine only 'like' surds.
To rationalize a denominator, multiply the numerator and denominator by the conjugate if it's a binomial surd.
Convert between exponential and logarithmic forms by identifying the base, exponent, and result.
Use logarithm laws to simplify expressions before evaluating.
When solving equations involving indices or logarithms, aim to isolate the variable using inverse operations and the properties of these functions.

Formulas

Product Rule: $a^m \times a^n = a^{m+n}$
Quotient Rule: $a^m \div a^n = a^{m-n}$
Power of a Power Rule: $(a^m)^n = a^{mn}$
Zero Exponent Rule: $a^0 = 1$
Negative Exponent Rule: $a^{-n} = \frac{1}{a^n}$
Fractional Exponent Rule: $a^{\frac{m}{n}} = \sqrt[n]{a^m} = (\sqrt[n]{a})^m$
Simplifying Surds: $\sqrt{ab} = \sqrt{a} \times \sqrt{b}$
Adding/Subtracting Surds: Only 'like' surds (same root) can be combined.
Multiplying Surds: $a\sqrt{b} \times c\sqrt{d} = ac\sqrt{bd}$
Dividing Surds: $\frac{a\sqrt{b}}{c\sqrt{d}} = \frac{a}{c}\sqrt{\frac{b}{d}}$
Rationalizing Denominators: $\frac{a}{\sqrt{b}} = \frac{a\sqrt{b}}{b}$
Rationalizing Denominators (Binomial): $\frac{a}{\sqrt{b}+\sqrt{c}} = \frac{a(\sqrt{b}-\sqrt{c})}{b-c}$
Exponential to Logarithmic Form: $a^x = y \iff \log_a(y) = x$
Logarithm Addition Law: $\log_a(x) + \log_a(y) = \log_a(xy)$
Logarithm Subtraction Law: $\log_a(x) - \log_a(y) = \log_a(\frac{x}{y})$
Logarithm Power Law: $\log_a(x^n) = n \log_a(x)$
Logarithm of 1: $\log_a(1) = 0$
Logarithm of Base: $\log_a(a) = 1$
Logarithm of Reciprocal: $\log_a(\frac{1}{x}) = -\log_a(x)$
Logarithm of Base Power: $\log_a(a^x) = x$

Key Terms

Index (Exponent): A number or symbol written above and to the right of another number or symbol, indicating the number of times the base is multiplied by itself.
Surd: An irrational number that is expressed using a root sign (e.g., √2, ³√5).
Rational Number: A number that can be expressed as a fraction p/q, where p and q are integers and q is not zero. Includes terminating and repeating decimals.
Irrational Number: A number that cannot be expressed as a simple fraction; its decimal representation is non-terminating and non-recurring.
Logarithm: The power to which a base must be raised to produce a given number.
Base (of a logarithm): The number that is raised to a power in an exponential expression or the number whose logarithm is being taken.
Perfect Square: A number that is the square of an integer (e.g., 4, 9, 16, 25).
Like Surds: Surds that have the same number under the radical sign (e.g., 3√5 and 7√5).