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Exponential and Logarithmic Functions. 3. Contents. 3.1 Rational Indices. 3.2 Logarithmic Functions. 3.3 Graphs of Exponential and Logarithmic Functions. 3.4 Applications of Logarithms.
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Exponential and Logarithmic Functions 3 Contents 3.1 Rational Indices 3.2 Logarithmic Functions 3.3 Graphs of Exponential and Logarithmic Functions 3.4 Applications of Logarithms
For a positive integer n, if , then x is and n th root of y, denoted by the radical . However, if n is even and y > 0, then or is the solution of the equation xn = y. If , then is usually written as 3.1 Rational Indices A. Radicals But in this chapter, we shall only consider the positive value of x. Remarks: • For n = 2, we call x the square root of y. • For n = 3, we call x the cube root of y.
3.1 Rational Indices For , we define the rational indices as follows: B. Rational Indices The laws of indices are also true for rational indices. where m, n are integers and n > 0.
For equation where b is a non-zero constant, p and q are integers with , we take the power of on both sides, 3.1 Rational Indices C. Using the Laws of Indices to Solve Equations
If a number y can be expressed in the form ax, where a > 0 and , It is denoted by If , then , where a > 0 and 3.2 Logarithmic Functions A. Introduction to Common Logarithm then x is called the logarithm of the number y to the base a.
3.2 Logarithmic Functions • If , then is undefined. (a) (b) (c) (d) (e) Notes : • When a = 10 (thus base 10), we write log x for log10x. This is called the common logarithm. • By the definition of logarithm and the laws of indices, we can obtain the following results directly.
The function , for x > 0 is called a logarithmic function. 1. 2. 3. 3.2 Logarithmic Functions B. Basic Properties of Logarithmic Functions Properties of Logarithmic functions: For M, N > 0,
3.2 Logarithmic Functions C. Using Logarithms to Solve Equations (a) Logarithmic Equations Logarithmic equations are the equations containing the logarithm of one or more variables. For example, log x = 2 is a logarithmic equation. In order to solve these kinds of equations, we need to use the definition and the properties of logarithm. For example, if log x = 2, then
3.2 Logarithmic Functions Exponential equations are the equations in the form ax = b, where a and b are non-zero constants and (b) Exponential Equations For such equations, we take logarithm on both sides and reduce the exponential equation to a linear equation, that is,
For bases other than 10, such as the function for and , they are also called logarithmic functions. 1. 2. 3. 4. 5. 3.2 Logarithmic Functions D. Other Types of Logarithmic Functions The logarithmic functions with different bases still have the following properties:
For a> 0 and , a function y = ax is called exponential function, where a is the base and x is the exponent. The following diagram shows the graph of for –3 x 3. 3.3 Graphs of Exponential and Logarithmic Functions A. Graphs of Exponential Functions Fig. 3.2
3.3 Graphs of Exponential and Logarithmic Functions Properties of the graph of exponential function: Fig. 3.2 • y = ax and y = a–x are reflectionally symmetric about the y-axis. • The graph does not cut the x-axis (that is y > 0 for all values of x). • The y-intercept is 1. 4. For the graph of y = ax, (a) if a > 1, then y increases as x increases; (b) if 0 < a < 1, then y decreases as x increases.
3.3 Graphs of Exponential and Logarithmic Functions B. Graphs of Logarithmic Functions Fig. 3.5 shows the graph of y = log x. Fig. 3.5
3.3 Graphs of Exponential and Logarithmic Functions • The function is undefined for The function f (x) = 10x is called the inverse function of the common logarithmic function f (x) = log x. Properties of the graph of logarithmic function: • For the graph of y = logx, • (a) x-intercept is 1; • (b) it does not have y-intercept; • (c) y increases as x increases. Fig. 3.5
3.4 Applications of Logarithms A. Transforming Data from Exponential Form to Linear Form We can actually transform data from exponential form to linear form. Suppose y = kxn, where k > 0 and n 0. Taking logarithm on both sides, we have which is a linear function with Y = log y, X = log x, a = log k and b = n.
3.4 Applications of Logarithms B. Applications of Logarithms in Real-life Problems • Loudness of Sound Decibel (dB) is the unit for measuring the loudness L of sound, which is defined as where I is the intensity of sound and I0 is the threshold of hearing for a normal person. Notes: I0 is the minimum audible sound intensity which is about 1012 W/m2.
3.4 Applications of Logarithms log E = 4.8 + 1.5R • Richter Scale The Richter scale (R) is a scale for measuring the magnitude of an earthquake. It is calculated from the energy E released from an earthquake and is given by the following formula, where E is measured in joules (J).
