Consider xf x a
This is a one-to-one function, therefore it has an inverse.
The inverse is called a logarithm function.
Example:416 2 24 log 16 Two raised to what power
is 16?
The most commonly used bases for logs are 10: 10log logx x
and e: log lne x x
lny x is called the natural log function.
logy x is called the common log function.
The log to the base “b” of “x” is the exponent to which “b” must
be raised to obtain “x”
y = log10 x
y = log e x
x = 10 y
x = e y
Properties of Logarithmic FunctionsIf b, M, and N are positive real numbers, b 1, and p and x are real numbers, then: Log15 1 = 0
Log10 10 = 1
Log5 5x = x
3log x = x 3
150 = 1
101 = 10
5x = 5x
The Decibel Scale
The decibel level D of a sound of intensity I , measured in watts per
square meter (W/ m2) is given by
where I0 = 10–12 W/ m2 is the intensity of the least audible sound that
an average healthy person can hear.
Sound Intensity, W/ m2 Sound
1.0 10–12 Threshold of hearing
5.2 10–10 Whisper
3.2 10–6 Normal conversation
8.5 10–4 Heavy traffic
3.2 10–3 Jackhammer
1.0 100 Threshold of pain
8.3 102 Jet plane with afterburner
0
log10DI
I
The magnitude M on the Richter scale of an earthquake that releases energy E , measured in joules, is given by
where E0 = 104.40 joules is the energy released by a small reference
earthquake.
0EE
log32
M
Magnitude on Richter scale Destructive power
M < 4.5 Small4.5 < M < 5.5 Moderate5.5 < M < 6.5 Large6.5 < M < 7.5 Major7.5 < M Greatest
The Richter Scale
Since logs and exponentials are inverses the domain and range switch!…the x values and y values are exchanged…
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