12.4 - Logarithmic functions

Click here to review functions. Click here to see how logarithmic functions compare with other types of functions.

Logarithmic functions are often used to describe quantities that vary over immense ranges. Large ranges of numbers occur in many settings: the amplification abilities of different electronic amplifiers, the sensitivity range of the human eye or the human ear, or the range of energies released by earthquakes. Logarithmic scales such as the decibel scale and the Richter scale are designed to describe large ranges of numbers.

Any logarithmic function can be expressed in the form:
y = a ln (x) + b,
where x and y are variables and a and b are constants. An example is:

y = 2.79 ln (x) + 5.80.
This function can be expressed in many equivalent forms using the change of base formula. For example:

y = 2.79 ln (x) + 5.80

= 6.43 log (x) + 5.80

= 5 log 6 (8 x).

The Graph of the Logarithmic Function

Here are graphs of the functions y = log (x) and y = ln (x).

Notice that immense variations in x correspond to small changes in y (at least in the region x > 1). Notice also that log(0) = −∞, and that logarithmic functions exist only to the right of the y axis. Thus the logarithm of a negative number does not exist. The graph of any logarithmic function has the same shape as these and will differ only in a vertical shift and/or a vertical stretch.

The Electromagnetic Spectrum

The electromagnetic spectrum is mentioned here because it is a nice example of a logarithmic scale.

Electromagnetic (EM) waves can be produced and detected in a variety or spectrum of wavelengths. The table below shows the EM spectrum and the names given to EM radiation of various wavelengths. The spectrum is actually infinite - we have simply shown the range from 10 −15m to 10 6m that has been intensively exploited by mankind so far.

The scale used to present the wavelengths is called a logarithmic scale. In this type of scale each factor of 10 in wavelength is allocated one line of the table. That is, the range from 10 1 to 10 2 is allocated the same amount of space as the range from 10 2 to 10 3, namely 1 line. By contrast in a linear scale the range from 10 2 to 10 3 (a range of 900) would be allocated 10 times as much space as the range from 10 1 to 10 2 (a range of 90).

wavelength (in meters)    name of region of EM spectrum
 10 −15  
 10 −14  
 10 −13  gamma
 10 −12  
 10 −11  
 10 −10  x-ray
 10 −9  
 10 −8  
 10 −7  ultraviolet
 10 −6  visible region
 10 −5  infrared
 10 −4  
 10 −3  
 10 −2  microwave
 10 −1  
 10 0  short radio waves
 10 1  FM radio
 10 2  AM radio
 10 3  
 10 4  
 10 5  long radio waves
 10 6  

Here is a brief history of the electromagnetic spectrum and the significance of some of its regions. In 1820 Oerstad and Ampere discovered that any wire carrying an electric current was surrounded by a magnetic field. A practical application of this effect was the electromagnet. In 1831 Michael Faraday discovered that a changing magnetic field near a wire could induce a voltage in the wire. A practical application of this effect was the electric generator.

In 1865 James Clerk Maxwell was able to combine the mathematical equations describing these two effects and to theoretically predict the existence of waves of electricity and magnetism, which we now call electromagnetic (EM) radiation. His equations also predicted the velocity of these waves. You can imagine his excitement when he discovered that their speed was exactly the speed of light! Twenty-two years later Hertz first produced and detected electromagnetic waves, and ten years after that Marconi invented the radio.

Our eyes are sensitive only to a very small part of the EM spectrum called the visible region. Different wavelengths in this region are perceived by us as the different colors of the rainbow. Visible light is caused by transitions of the outermost electrons of atoms from one orbit to another.

We feel as heat on our skin (rather than see) the infrared region of the EM spectrum. And our skin becomes tanned by ultraviolet radiation.

EM radiation from the microwave region causes the water molecule to vibrate. This effect can be used to cook food in a microwave oven. (In case you wondered, the metal screen on the oven door has a fine enough mesh to short circuit the electric component of the microwaves so they can't escape from the oven but is coarse enough to let the shorter wavelength visible light through so that we can see what's cooking inside.)

X-rays are emitted by the inner electrons of atoms as they fall from a higher orbit to a lower one. Gamma rays are caused by the rearrangement of particles inside an unstable atomic nucleus.

The energy of radiation depends on two things: the intensity (brightness) of the radiation, and the wavelength; short wavelengths being more energetic than long wavelengths. Thus gamma rays are the most energetic and dangerous form of radiation.

Astronomy is one area of science that uses every part of the EM spectrum. Radio telescopes as big as several kilometers across gather the long-wavelength radiation emitted by molecules in interstellar space. X-ray telescopes (launched into earth orbit to avoid absorption of x-rays by the earth's atmosphere) observe short-wavelength radiation emitted by black holes, neutron stars and other exotic objects.

The Decibel Power Scale

An ideal amplifier multiplies the height of an input signal by some factor and outputs it without distortion.

If the height of the signal represents power then the amplification factor is the ratio Pout /Pin where Pout is the output power and Pin is the input power, both measured in Watts at the same instant in time. For lasers the ratio can be up to 10 12. Attenuators, of which microphones are an example, are devices for which the ratio Pout /Pin is much less than 1. To compress these large variations we define the decibel scale:
N is called the gain of the amplifier or attenuator. It is measured in units called decibels (abbreviated db). Notice that the gain is positive for amplifiers, negative for attenuators and zero if Pout /Pin = 1.

Example: What is the output power Pout of an amplifier with a gain of 40 db if Pin = 50 mW?

Solution: Substitute the given values into the gain formula and then solve for Pout :

Amplifiers in Cascade: To achieve very large amplification factors amplifiers are sometimes cascaded, which means that the output of one amplifier becomes the input of the next amplifier:

The amplification factor of the whole system is the product of the amplification factors of the components. The whole system in the figure has an amplification factor of 6. Equivalently, the gain of the system equals the sum of the gains of its components. To prove this statement let Pin be the power going into amplifier # 1, Pout be the power coming out of amplifier # 2 and Pint be the intermediate power (coming out of Amplifier # 1 and going into # 2). The gain of the whole system is:
Now put a factor of Pint /Pint inside the brackets:
Now use property 1 of logarithms:
The first term in the final expression is just the gain of amplifier # 1 and the second term is the gain of amplifier # 2. This proves that the gain of a system equals the sum of the gains of its components.

Example: Find the output power of the system shown:

Solution: The gain of the whole system is N = 45 db. Therefore:
45 = 10 log 10 (Pout / 0.05W)
Solving for Pout gives:
Pout = 0.05 W · 10 4.5 = 1581 W

Sometimes the gain formula is used to compare the signal power to the noise power. For example a signal-to-noise ratio of 1000:1 may be quoted as +30 db.

The Decibel Loudness Scale

Fluctuations in air pressure cause our eardrums to vibrate and we interpret these vibrations as sound. The loudness L of a sound (measured in decibels) is defined as:
where P is the pressure fluctuation of the sound and Po is the pressure fluctuation of a sound at the threshold of human hearing, which is 20 μPa.

Example: How many times larger are the pressure fluctuations of average traffic noise (with a loudness of 85 db) than average livingroom noise (with a loudness of 40 db)?

Solution: Substituting each noise into the loudness formula, we have:
We can eliminate Po and compare Ptraffic and Proom directly by subtracting these equations and using property 2 of logarithms:
Solving for the ratio Ptraffic /Proom gives:

The acoustic loudness scale can also be defined in terms of the intensity I of the sound (the energy/unit area/unit time falling on the eardrum):
The threshold intensity for human hearing is Io = 10 −12 Watts / m2. A factor of 10 appears in this formula (rather than 20) since the energy of a wave is proportional to the square of the pressure fluctuation.

The Richter Earthquake Scale

The Richter number R describing the strength of an earthquake is defined as:
where E is the energy released by the earthquake and Eo is a reference arbitrarily set at the limit of sensitivity of Richter's original seismic measuring apparatus.

The pH Acidity Scale

The pH number of an acid or base is defined as:
where [ H + ] is the concentration in moles/liter of the H +  ion responsible for acidity. A solution with pH = 7 is neutral, pH < 7 is an acid and pH > 7 is a base.

The Astronomical Brightness Scale

The magnitude M of a star describes its brightness as it appears to us on earth and is defined as:
where I is the intensity of the light from the star. I1 is the intensity of a first magnitude star (the 20 brightest stars in the sky are approximately magnitude 1). The most powerful telescopes can detect stars as faint as magnitude +24.