Chapter 3 Exponential And Logarithmic Functions

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Exponential and Logarithmic Functions 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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3.1 Rational Indices A. Radicals n For a positive integer n, if x = y , then x is and n th root of y, denoted by the . radical n y

However, if n is even and y > 0, then x = + n y or − n y is the solution of the equation xn = y. But in this chapter, we shall only consider the positive value of x. If x n = y , then x = n y . Remarks: Content

•

For n = 2, we call x the square root of y. For n = 3, we call x the cube root of y.

1.

2

y is usually written as

y.

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3.1 Rational Indices B. Rational Indices The laws of indices are also true for rational indices. For y ≠ 0 , we define the rational indices as follows: 1 n

y =n y m n

y = (n y ) m = n y m where m, n are integers and n > 0. Content

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3.1 Rational Indices C. Using the Laws of Indices to Solve Equations p q

For equation x = b where b is a non-zero constant, p and q are integers with q q ≠ 0 , we take the power of p on both sides, p q

q p

q p

p q × q p

q p

(x ) = b x Content

∴

=b

x=b

q p

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3.2 Logarithmic Functions A. Introduction to Common Logarithm If a number y can be expressed in the form ax, where a > 0 and a ≠ 1 , then x is called the logarithm of the number y to the base a. It is denoted by x = log a y.

If y = a x , then log a y = x, where a > 0 and a ≠ 1. Content

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3.2 Logarithmic Functions Notes :

Content

1.

If y ≤ 0 , then log a y is undefined.

•

When a = 10 (thus base 10), we write log x for log10 x. This is called the common logarithm.

1.

By the definition of logarithm and the laws of indices, we can obtain the following results directly. (a)

1 = 10 0 ,

∴

log1 = 0

(b)

10 = 101 ,

∴

log10 = 1

(c)

100 = 10 2 ,

∴

log100 = 2

∴

log 0.1 = −1

(d)

0.1 = 10 −1 ,

(e)

0.01 = 10 −2 ,

∴ log 0.01 = −2

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3.2 Logarithmic Functions B. Basic Properties of Logarithmic Functions The function f ( x) = log x , for x > 0 is called a logarithmic function. Properties of Logarithmic functions:

For M, N > 0, 1. log(MN ) = log M + log N 2. log Content

M = log M − log N N

n 3. log M = n log M

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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. Content

For example, if log x = 2, then x = 10 2 = 100

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3.2 Logarithmic Functions (b) Exponential Equations Exponential equations are the equations in the form ax = b, where a and b are non-zero constants and a ≠ 1. For such equations, we take logarithm on both sides and reduce the exponential equation to a linear equation, that is, log a x = log b x log a = log b Content

x=

log b log a

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3.2 Logarithmic Functions D. Other Types of Logarithmic Functions For bases other than 10, such as the function f ( x) = log a x for x > 0, a > 0 and a ≠ 1, they are also called logarithmic functions. The logarithmic functions with different bases still have the following properties:

1.

log a a = 1

2. log a 1 = 0

3.

log a ( MN ) = log a M + log a N

4. log a

5.

log a M n = n log a M

Content

M = log a M − log a N N

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3.3 Graphs of Exponential and Logarithmic Functions A. Graphs of Exponential Functions For a > 0 and a ≠ 1 , a function y = ax is called exponential function, where a is the base and x is the exponent. x −x The following diagram shows the graph of y = 2 and y = 2 for –3 ≤ x ≤ 3.

Content

Fig. 3.2

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3.3 Graphs of Exponential and Logarithmic Functions Properties of the graph of exponential function:

Fig. 3.2

Content

•

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.

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3.3 Graphs of Exponential and Logarithmic Functions B. Graphs of Logarithmic Functions Fig. 3.5 shows the graph of y = log x.

Content

Fig. 3.5

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3.3 Graphs of Exponential and Logarithmic Functions 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:

Content

1.

The function is undefined for x ≤ 0.

•

For the graph of y = log x, (a)

x-intercept is 1;

(b)

it does not have y-intercept;

(c)

y increases as x increases.

Fig. 3.5

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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 log y = log(kx n ) log y = log k + n log x Y = a + bX , Content

which is a linear function with Y = log y, X = log x, a = log k and b = n.

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3.4 Applications of Logarithms B. Applications of Logarithms in Real-life Problems 1.

Loudness of Sound Decibel (dB) is the unit for measuring the loudness L of sound, which is defined as L = 10 log

I , I0

where I is the intensity of sound and I0 is the threshold of hearing for a normal person. Content

Notes: I0 is the minimum audible sound intensity which is about 10−12 W/m2.

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3.4 Applications of Logarithms 1.

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, log E = 4.8 + 1.5R where E is measured in joules (J).

Content

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