IB Mathematics Analysis and Approaches Topic SL 1.7—laws of Exponents and Logs Revision Notes (SL/HL)

IB Mathematics Analysis and Approaches Topic SL 1.7—laws of Exponents and Logs Revision Notes (SL/HL) | RevisionDojo

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Laws of Exponents with Rational Exponents

Rational Exponents and Roots

For any positive real number $a$ and rational number $\frac{m}{n}$ where $m$ is an integer and $n$ is a positive integer:

$$a^{\frac{m}{n}} = \sqrt[n]{a^m}$$

Example

Let's evaluate $8^{\frac{2}{3}}$:

$$8^{\frac{2}{3}} = \sqrt[3]{8^2} = \sqrt[3]{64} = 4$$

Here, we first square 8, then take the cube root of the result.

Note

When $n$ is even, $a^{\frac{1}{n}}$ refers specifically to the positive $n$th root of $a$.
This convention ensures that exponential expressions with rational exponents are well-defined for positive real bases.

Properties of Exponents

The laws of exponents are as follows:
1. Product rule: $a^x \cdot a^y = a^{x+y}$
2. Quotient rule (for $a \neq 0$): $\frac{a^x}{a^y} = a^{x-y}$
3. Power rule: $(a^x)^y = a^{xy}$
These rules apply regardless of whether $x$ and $y$ are integers, rational, or real numbers.

Example question

Evaluate $27^{\frac{4}{3}}$.

Solution

$$27^{\frac{4}{3}} = 27^{\frac{1}{3} \cdot 4} = \sqrt[3]{27}^4 = 3^4 = 81$$

We applied the power rule, then simplified the rational exponent, and finally evaluated the result.

Common Mistake

Students often confuse $a^{\frac{m}{n}}$ with $\frac{a^m}{a^n}$.
Remember, $a^{\frac{m}{n}}$ is a single number raised to a fractional power, while $\frac{a^m}{a^n}$ is a quotient of two exponential expressions that evaluates to $a^{m - n}$.

Laws of Logarithms

Logarithms are the inverse operations of exponents, and they follow a set of laws that mirror the properties of exponents.

Basic Logarithm Laws

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Laws of Exponents with Rational Exponents

Rational Exponents and Roots

For any positive real number $a$ and rational number $\frac{m}{n}$ where $m$ is an integer and $n$ is a positive integer:

$$a^{\frac{m}{n}} = \sqrt[n]{a^m}$$

Example

Let's evaluate $8^{\frac{2}{3}}$:

$$8^{\frac{2}{3}} = \sqrt[3]{8^2} = \sqrt[3]{64} = 4$$

Here, we first square 8, then take the cube root of the result.

Note

When $n$ is even, $a^{\frac{1}{n}}$ refers specifically to the positive $n$th root of $a$.
This convention ensures that exponential expressions with rational exponents are well-defined for positive real bases.

Properties of Exponents

The laws of exponents are as follows:
1. Product rule: $a^x \cdot a^y = a^{x+y}$
2. Quotient rule (for $a \neq 0$): $\frac{a^x}{a^y} = a^{x-y}$
3. Power rule: $(a^x)^y = a^{xy}$
These rules apply regardless of whether $x$ and $y$ are integers, rational, or real numbers.

Example question

Evaluate $27^{\frac{4}{3}}$.

Solution

$$27^{\frac{4}{3}} = 27^{\frac{1}{3} \cdot 4} = \sqrt[3]{27}^4 = 3^4 = 81$$

We applied the power rule, then simplified the rational exponent, and finally evaluated the result.

Common Mistake

Students often confuse $a^{\frac{m}{n}}$ with $\frac{a^m}{a^n}$.
Remember, $a^{\frac{m}{n}}$ is a single number raised to a fractional power, while $\frac{a^m}{a^n}$ is a quotient of two exponential expressions that evaluates to $a^{m - n}$.

Laws of Logarithms

Logarithms are the inverse operations of exponents, and they follow a set of laws that mirror the properties of exponents.

Basic Logarithm Laws

Definition

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Note

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Excepteur sint occaecat cupidatat non proident

Tip

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Sed do eiusmod tempor incididunt ut labore et dolore magna aliqua.
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Duis aute irure dolor in reprehenderit in voluptate velit esse cillum.

Number and Algebra16 subtopics

Functions16 subtopics

Geometry & Trigonometry18 subtopics

Statistics & Probability14 subtopics

Calculus19 subtopics

IB Mathematics Analysis and Approaches SL 1.7—laws of Exponents and Logs Notes

Number and Algebra16 subtopics

Functions16 subtopics

Geometry & Trigonometry18 subtopics

Statistics & Probability14 subtopics

Calculus19 subtopics

Laws of Exponents with Rational Exponents

Rational Exponents and Roots

Properties of Exponents

Laws of Logarithms

Basic Logarithm Laws

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Laws of Exponents with Rational Exponents

Rational Exponents and Roots

Properties of Exponents

Laws of Logarithms

Basic Logarithm Laws

anim nostrud sit dolore minim proident quis fugiat velit et eiusmod nulla quis nulla mollit dolor sunt culpa aliqua

Duis aute irure dolor in reprehenderit

Excepteur sint occaecat cupidatat non proident

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