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Graph Properties

1.4 Logarithmic Functions

The Logarithmic Function

  • The logarithmic function with base is defined as follows:

a^{x} = b is equivalent to log_{a}{b}=x, where a \in R^{+}\setminus \left \{ 1 \right \} .

Note:a \in R^{+}\setminus \left \{ 1 \right \}‘ means that could be any positive number, excluding 1.

  • The expression log_{a}{y}=x is read as: ‘the logarithm of y to the base a is equal to x’.

Example

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1.3 Exponential Functions [Free]

The Exponential Function

  • The exponential function is defined as follows:

y=a^x, where a \in R^+\backslash {1}.

Note:a \in R^+\backslash {1}’ means that could be any positive number, excluding 1.

Example

9=3^{2}=3 \times 3 is equivalent to log_{3}9=2.

8=16^{\frac{3}{4}}=\sqrt[4]{16^{3}}=\sqrt{\sqrt{16 \times 16 \times 16}}.

Graph & Properties

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1.2 Power Functions

Introduction to Power Functions

  • Power functions are functions with the form f(x)=x^{n}, where n is a rational number.
  • Some examples of power functions are:

f(x)=x^{2},\ f(x)=x^{4},\ f(x)=x^{\frac{1}{4}},\ f(x)=x^{-5},\ f(x)=x^{\frac{1}{3}}

  • The expression x^n is read as: ‘x to the power of n’.

Drawing Power Functions (for Positive Integers) and its Graph Properties

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