The logarithmic function with base 10 is called the common logarithm.

We write as y = log x

A calculator with a log key can be used to find common logarithms of any positive number.

One mathematical model that contains a common logarithm is the formula for pH of a solution:

where is the hydronium ion concentration in moles per liter. pH measures the acidity or alkalinity of solutions. An acid solution has pH < 7; solutions with pH > 7 are called basic. Water has pH = 7.

The hydronium concentration of milk is 3.97 x 10. To find the pH of milk:

	pH formula
	Substitution
pH = - [ -6.4] pH = 6.4

The pH of milk is about 6.4.

The magnitude R on the Richter scale of an earthquake of intensity I is given by

where is the intensity of a zero-level earthquake.

Northern California’s 1989 earthquake was 10 times as intense as a zero-level earthquake; that is, To find the magnitude on the Richter scale:

	Formula for magnitude
	Substitute for I
R = log 10	Simplify
R = 7.1	Logarithm Property 6

Northern California’s 1989 earthquake registered 7.1 on the Richter scale.

The model that describes sound intensity is similar to the model for earthquake intensity. The loudness of sounds is measured in decibels. The loudness level of a sound, d, in decibels, is given by

where I is the intensity of the sound and is the intensity of a sound barely audible to the human ear.

The cry of a blue whale can be heard nearly 500 miles away, reaching an intensity of

. To determine the decibel level of this sound:

	Loudness formula
	Substitute the given intensity for I
d = 10 log (6.3 x 10)	Simplify
d = 10(18.799)	Find the logarithm
d = 187.99

The decibel level of the blue whale’s cry is 187.99.

The logarithmic function with base e is called the natural logarithmic function. We write:

Natural logarithms can be found with a calculator that has an ln key.

Calculators give the values of both common logarithms (base 10) and natural logarithms (base e). To find a logarithm with any other base, we can use the Change-of-Base Theorem:

This theorem is used to write a logarithm in terms of quantities that can be evaluated with a calculator. Because calculators have keys for common and natural logarithms, we will change the given base, a, to either base 10 or base e. If we change the base to 10, the theorem looks like this:

b = 10;

If we change the base to e, we have:

b = e;

Example. Use the Change-of-Base Theorem to find each logarithm to the nearest hundredth: a. b.

Solution.

a.

	Change of base formula with base 10. Either base 10 or e can be used.
	Substitution
» 2.79	Evaluate logs; round to the nearest hundredth

b.

	Change of base formula with base e. Again, either base 10 or e can be used.
	Substitution
» 5.61	Evaluate; round to hundredths