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0.9 Logarithms

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This module is part of a collection of modules intended for use by preengineering students enrolled in MATH 108 at the University of Texas at El Paso. This module addresses some applications of logarithms in several fields of engineering. Examples are presented.

Logarithms

Introduction

This module is intended to present some areas of engineering in which logarithms are used. By reading the material and solving the associated problems, you will learn about some important applications of logarithms in engineering.

Decibels

The decibel ( dB ) is a logarithmic unit that indicates the ratio of a physical quantity relative to a specified or implied reference level. The decibel is used for a wide variety of measurements in science and engineering, most prominently in acoustics, electronics, communications, radar, sonar and control systems.

Decibels are frequently used as a means to express the power ratio for physical systems. It is computed by multiplying the factor 10 by the base 10 logarithm of the ratio of the quantities under consideration. Equation (1) shows the computation that is used to express the ratio of two powers using decibels

L DB = 10 log 10 P 2 P 1 size 12{L rSub { size 8{ ital "DB"} } ="10"`"log" rSub { size 8{"10"} } left ( { {P rSub { size 8{2} } } over {P rSub { size 8{1} } } } right )} {}

Gain of an Amplifier: We will begin our discussion of decibels with an application in the field of electronics. An amplifier is an electronic device that is capable of boosting the power present in an input signal to produce an output signal with more power. It can be thought of as a black box as shown in Figure 1.

Block diagram of an amplifier.

In practical cases, the ratio of the power in the output signal to the power in the input signal is a positive quantity whose value is greater than unity. The decibel measurement of this ratio of power is often called the gain of the amplifier and is given as

Gain = 10 log 10 P output P input dB size 12{ ital "Gain"="10"`"log" rSub { size 8{"10"} } left ( { {P rSub { size 8{ ital "output"} } } over {P rSub { size 8{ ital "input"} } } } right )~ ital "dB"} {}

Question: An electronic signal is passed through an amplifier. Suppose that the power present in the signal at the input to the amplifier is 10 W. The power present in the signal at the output of the amplifier is 20 W. Express the gain of the amplifier in decibels.

We can use equation (2) to easily express the gain of the amplifier in terms of decibels

Gain = 10 log 10 20 W 10 W = 10 log 10 ( 2 ) = 3 . 01 dB ≈ 3 dB size 12{ ital "Gain"="10"`"log" rSub { size 8{"10"} } left ( { {"20"`W} over {"10"`W} } right )="10"`"log" rSub { size 8{"10"} } \( 2 \) =3 "." "01"` ital "dB" approx 3` ital "dB"} {}

Signal to noise ratio

Electrical signals are often corrupted by a random phenomenon known as noise when they are transmitted from one point to another . Because it is impossible to know the exact value of the noise at any point in time, it is often becomes difficult to extract the orignal signal at the receiver without the application of some form of signal processing algorithm such as a filter . The situation is depicted in Figure 2.

Communication system with signal processing.

A common figure of merit of communication systems is the signal-to-noise ratio . Communication systems that are characterized by high signal-to-noise ratios are in general superior to those that are characterized by low signal-to-noise ratios.

By definition the signal-to-noise ratio or SNR is given as the ratio of the power in a signal divided by the power in the noise that is responsible for corrupting the signal. The signal-to-noise ratio can be expressed in decibels as follows

Questions & Answers

A golfer on a fairway is 70 m away from the green, which sits below the level of the fairway by 20 m. If the golfer hits the ball at an angle of 40° with an initial speed of 20 m/s, how close to the green does she come?
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Samuel Reply
can someone explain to me, an ignorant high school student, why the trend of the graph doesn't follow the fact that the higher frequency a sound wave is, the more power it is, hence, making me think the phons output would follow this general trend?
Joseph Reply
Nevermind i just realied that the graph is the phons output for a person with normal hearing and not just the phons output of the sound waves power, I should read the entire thing next time
Joseph
Follow up question, does anyone know where I can find a graph that accuretly depicts the actual relative "power" output of sound over its frequency instead of just humans hearing
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"Generation of electrical energy from sound energy | IEEE Conference Publication | IEEE Xplore" ***ieeexplore.ieee.org/document/7150687?reload=true
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A string is 3.00 m long with a mass of 5.00 g. The string is held taut with a tension of 500.00 N applied to the string. A pulse is sent down the string. How long does it take the pulse to travel the 3.00 m of the string?
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Source:  OpenStax, Math 1508 (laboratory) engineering applications of precalculus. OpenStax CNX. Aug 24, 2011 Download for free at http://cnx.org/content/col11337/1.3
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