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International Baccalaureate Mathematics

Number and Algebra

Syllabus Content

Laws of exponents with integer exponents. Introduction to logarithms with base 10 and e. Numerical evaluation of logarithms using technology

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Examples

$$5^3 \times 5^{-6} = 5^{-3}$$ $$ 6^4 \div 6^3 = 6$$ $$ (2^3)^{-4} = 2^{-12}$$ $$ (2x)^4 = 16x^4$$ $$ 2x^{-3} = \frac{2}{x^3}$$

Awareness that \(a^x=b\) is equivalent to \(log_ab=x\), that \(b \gt 0\), and \( log_ex=lnx\).

Formula Booklet:

Exponents and logarithms

\( a^x = b \iff log_ab\), where \( a \gt 0, b \gt 0, a \neq 1\)

The laws of exponents, also known as the rules of indices, provide a set of rules for simplifying expressions involving powers or exponents. When working with integer exponents, these laws help to manipulate and simplify algebraic expressions. The key formulae for the laws of exponents with integer exponents are as follows:

$$ a^m \times a^n = a^{m+n} \\ a^m \div a^n = a^{m-n} \\ (a^m)^n = a^{m \times n} \\ a^0 = 1 \quad (\text{where } a \neq 0) \\ a^{-n} = \frac{1}{a^n} \quad (\text{where } a \neq 0) $$

In the formulae above, \( a \) is the base and \( m \) and \( n \) are integer exponents.

Logarithms are a mathematical concept that provides a way to work with exponential equations in a more linear manner. The logarithm of a number to a given base is the power or exponent to which the base must be raised to produce that number. The most commonly used bases in logarithms are 10, often referred to as the common logarithm, and \( e \), the base of natural logarithms. The number \( e \) is an irrational number approximately equal to 2.71828 and has profound significance in mathematics, especially in calculus and complex analysis.

The key formulae for logarithms with base 10 and \( e \) are:

$$ \log_{10}(x) = y \quad \text{means} \quad 10^y = x \\ \log_e(x) = \ln(x) = y \quad \text{means} \quad e^y = x \\ \log_b(m \times n) = \log_b(m) + \log_b(n) \\ \log_b\left(\frac{m}{n}\right) = \log_b(m) - \log_b(n) \\ \log_b(m^n) = n \times \log_b(m) $$

This video on Exponential and Logarithmic Functions is from Revision Village and is aimed at students taking the IB Maths AA Standard level course.

This video on Log Laws and Solving Equations is from Revision Village and is aimed at students taking the IB Maths AA Standard level course

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