Indices and Logarithms

Indices

• An number in index form is the form b^n
  • We call b the base
  • We call n the index/power/exponent
• Index notation is useful for writing large or small numbers in a manageable form

Exponential Functions

• A function in the form f(x)=a^x, where a is positive is called an exponential function

• For a>1 function grows as x increases

• All such curves pass through (0,1) since a^0 = 1(for a>1)

• For 0<a<1 function decays as x increases

• All such curves pass through (0,1) since a^0 = 1(for 0<a<1)

• The following function : f(x)=e^x, is called the natural exponential function.

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Indices and Logarithms

Rules of Indices

Law 1:	$a^p \cdot a^q = a^{p+q}$	Law 4:	$a^0 = 1$	Law 7:	$a^{-p} = \frac{1}{a^p}$
Law 2:	$\frac{a^p}{a^q} = a^{p-q}$	Law 5:	$a^{\frac{1}{n}} = \sqrt[n]{a}$	Law 8:	$(ab)^p = a^p b^p$
Law 3:	$(a^p)^q = a^{pq}$	Law 6:	$a^{\frac{p}{n}} = (\sqrt[n]{a})^p$	Law 9:	$(\frac{a}{b})^p = \frac{a^p}{b^p}$

• $\sqrt[n]{a}$ is called the nth root of a if b is the nth root of a, then $b^n = a$
• Ex: $\sqrt{a} = a^{1/2}$

With x as an index

• If $a^x = a^y$ and $a \neq -1, 0, 1$ then we can say x=y.

Surds

• Any number of the form $\sqrt[n]{a}$, n > 1, n ∈ N, which cannot be written as a fraction is called a surd

If a, b > 0 then:

Law 1:	$\sqrt{a}\sqrt{b} = \sqrt{ab}$	Law 2:	$\frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}}$

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Indices and Logarithms

Rationalising the denominator

• If a fraction has an irrational denominator we may need to rationalize the denominator
• The means multiplying the numerator and denominator by the conjugate of the denominator.
• The conjugate of $x+y$ is $x-y$ where $x,y \in \mathbb{R}$
• ***NB $(x+y)(x-y) = x^2 - y^2$

Logarithms

• Logarithms reverse the process of exponentiation
• $a^n = n \Leftrightarrow \log_a n = m$, where a (the base) > 0 and ∈ ℝ, N > 0

Graphing a Logarithmic Function

• The graph of $y = \log_a x$ is a reflection of the graph $y = 2^x$ in the line $x=y$
• ***NB we know $a^x = a^y \Leftrightarrow x = y$
• Therefore: if $\log_a x = \log_a y \Leftrightarrow x = y$

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Laws of Logarithms

Laws Overview

Law 1:	$\log_a(xy) = \log_a x + \log_a y$	Law 5:	$\log_a(\frac{1}{x}) = -\log_a x$
Law 2:	$\log_a(\frac{x}{y}) = \log_a x - \log_a y$	Law 6:	$\log_a(a^x) = x$
Law 3:	$\log_a(x^q) = q\log_a x$	Law 7:	$a^{\log_a x} = x$
Law 4:	$\log_a 1 = 0$	Law 8:	$\log_b x = \frac{\log_a x}{\log_a b}$

Key Points

• The natural logarithm function loge has the irrational number its base. It is also written as ln(x)
• Logarithms can be used to solve practical problems.