How Loud is Too Loud Activity

The decibel (dB) is the unit to measure the intensity of a sound. It is a logarithmic unit used to express the ratio between two values of a physical quantity. The smallest audible sound (near total silence) is 0 dB. A sound 10 times more powerful is 10 dB. A sound 100 times more powerful (102) is 20 dB.

In other words, when you increase sound (in terms of decibels) it is an exponential increase.

How Loud is Too Loud Activity

Audible wavelengths vary from animal to animal

Properties of Logarithms

What is a Logarithm?0A quantity representing the power to which a fixed

number (the base) must be raised to produce a given number

What is it used for?0Multiplying large numbers

without the use of a calculator0 Remember calculators were

not readily available to the masses until about 50 years ago. We’ve come a LONG way since then in technology.

Greek

Greek

Modern LatinLogos

Reckoning,Ratio

Arithmosnumber

Origin

logarithmus Logarithm(log)

Early 17th century

Why do we need Logarithms?0Calculators have advanced enough that even the most basic of

calculators can easily compute logarithms for us, however, it’s important that we still understand the use of logarithms.

0The exponential function y=ax is one of the most important functions in mathematics, physics, and engineering. Radioactive decay, bacterial growth, population growth, continuous interest, etc. are all exponential examples we use everyday.

0How do you solve for x if it is in the exponent? (y=ax)

0We need logarithms to help us solve.

y = ax is equivalent to loga(y) = x

ExamplesWrite the following in logarithmic/exponential form:

ExamplesEvaluate the expression

Different Bases:(We will practice logarithms using a variety of different bases.)

Below you will find a list of the most common forms.

0Base 10 is used by chemists in their measurements of pH, the acidity of a liquid. We often call this the common base.

0Base 2 is used in information theory and computers (used in transmitting information and measuring the errors made and how to correct those errors).

0Base e, where e=2.718281828..., is used in calculus and is probably the most important base.

Let’s Discover the Properties of Logarithms…

Here’s what you should have come up with…

Here’s what you should have come up with…

OR

Here’s a few others for your notes…

Let’s consider

1. Take the log of both sides2. Use the multiplication law

of logs to separate the right side of the equation into two separate logarithms

3. Use the exponent law of logs to simplify further, turning into 2 times log(a)

The reverse is also trueConsider

1. Start with the exponent law of logarithms

2. Use the division/subtraction law of logarithms

3. Simplify0 Note: We DID NOT DIVIDE BY LOG!

This is not a defined mathematical principle.

0 We concluded that both sides of the equation had log10 therefore their values are equivalent.

Let’s look at some real applications...

Decibels

A proposed city ordinance will make it illegal to create sound in a residential area that exceeds 72 decibels during the day and 55 decibels during the night. How many times as intense is the noise level allowed during the day than at night (Hint: is the equation for loudness, L, in decibels, when R is the relative intensity of the sound)

times greater

Earthquakes

The magnitude of an earthquake is measured on a logarithmic scale called the Richter scale. The magnitude M is given by , where x represents the amplitude of the seismic wave causing ground motion.

a) How many times as great is the amplitude caused by an earthquake with a Richter scale rating of 7 as an aftershock with a Richter scale rating of 4?

b) How many times as great was the motion caused by the 1906 San Francisco earthquake that measured 8.3 on the Richter scale as that caused by the 2001 Bhuj, India, earthquake that measured 6.9?

times greater

times greater