Lesson 7B: Evaluating Logarithmic Expressions

Sound is a wave in the air and the volume of the sound is related to the intensity of the wave.

	The intensity of a whisper is approximately 100.
	The intensity of background noise in a quiet rural area is approximately 1 000.
	The intensity of normal conversation is approximately 1 000 000.
	The intensity of a rock concert is approximately 1 000 000 000 000.

To compare these values, try to place them on a number line. The difficulty is that visualizing or graphing something going from one hundred to one trillion is almost impossible - the range is too great.

Fortunately, sound volume is usually not measured in intensity, but in loudness, which is defined by the formula L = 10 log₁₀ I, where L is the loudness (measured in decibel or dB) and I is the intensity. This formula is a logarithmic function. How does the use of a logarithmic function simplify the problem of placing the large intensity values on a number line?

To find out, use the logarithmic function L = 10 log₁₀ I to calculate the loudness, L, for each of the intensities on the previous page.

Whisper: L = 10 log₁₀ I = 10 log₁₀(100) = 20 dB

Rural setting: L = 10 log₁₀ I = 10 log₁₀(1 000) = 30 dB

Conversation: L = 10 log₁₀ I = 10 log₁₀(1 000 000) = 60 dB

Rock concert: L = 10 log₁₀ I = 10 log₁₀(1 000 000 000 000) = 120 dB

Using the new values for loudness rather than intensity, constructing a number line and comparing the values is easy.

This is called a logarithmic scale because the values compared are logarithmic values. To compare two values on this logarithmic scale, you must realize that each step up the number line, which is an increase in loudness of 10 decibels, represents a 10 times increase in intensity. Explore what this means by comparing each value of loudness on the number line to the loudness of a whisper.

A whisper is 20 dB.

A rural setting is 30 dB, one space from a whisper on the number line. The rural setting is 10 dB louder than a whisper; therefore, it is 10 times as intense.

A conversation is 60 dB, four spaces from a whisper on the number line. The conversation is 40 dB louder than a whisper. However, each space represents a 10 times increase in intensity. Therefore, the conversation is 10 × 10 × 10 × 10 = 10 000 times as intense as a whisper.

A rock concert is 120 dB, ten spaces from a whisper on the number line. The rock concert is 100 dB louder than a whisper. Because each space represents a 10 times increase in intensity, the rock concert is 10 × 10 × 10 × 10 × 10 × 10 × 10 × 10 × 10 × 10 = 10 000 000 000 times as intense as a whisper.

Often, logarithmic scales are used to display numeric values that represent natural phenomena. Examples include the Richter magnitude scale for strength of earthquakes and movement in the earth; bel and decibel for acoustic power (loudness) and electric power; minor second, major second, and octave for the relative pitch of notes in music; and pH for acidity and alkalinity.

In this Training Camp, you will simplify logarithmic expressions and equations and solve problems that involve logarithmic scales related to specific natural phenomena.

	express a logarithmic equation as an exponential equation and vice versa
	determine the approximate value of a logarithmic expression with and without technology
	solve problems that involve logarithmic scales, such as the Richter scale and the pH scale