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 (10 2 ) 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? 0 A quantity representing the power to which a fixed number (the base) must be raised to produce a given number What is it used for? 0 Multiplying large numbers without the use of a calculator 0 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 Modern Latin Logos Reckoning, Ratio Arithmos number Origin logarithmusLogarithm (log) Early 17 th century

Why do we need Logarithms? 0 Calculators 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. 0 The exponential function y=a x 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. 0 How do you solve for x if it is in the exponent? (y=a x ) 0 We need logarithms to help us solve. y = a x is equivalent to log a (y) = x

Examples Write the following in logarithmic/exponential form:

Examples Evaluate 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. 0 Base 10 is used by chemists in their measurements of pH, the acidity of a liquid. We often call this the common base. 0 Base 2 is used in information theory and computers (used in transmitting information and measuring the errors made and how to correct those errors). 0 Base e, where e= , 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 a few others for your notes…

1. Start with the exponent law of logarithms 2. Use the division/subtraction law of logarithms 3. Simplify 0 Note: We DID NOT DIVIDE BY LOG! This is not a defined mathematical principle. 0 We concluded that both sides of the equation had log 10 therefore their values are equivalent.

Let’s look at some real applications...

Decibels

Earthquakes