# Logarithms

## Where Logarithms Are Used

Logarithms occur in many every day problems. Architects and engineers use logarithms to model air flow in buildings. In finance logarithms are used to calculate the monthly payment required for a mortgage. The Richter Scale, used to tell the magnitude of an earthquake, is a logarithm based equation. That is why an earthquake of magnitude 7.2 is ten times bigger than one with a magnitude of 6.2.

In the old days, before the invention and availability of inexpensive calculators in the late 1960's, all math was done by hand. For complicated problems, the student would use a table of logarithms. Engineers and scientists who did lots of calculations used a device called a slide rule. It was based on the properties of logarithms and used Scientific Notation to simplify the numbers.

## Definition

Logarithms are the inverse operation to exponential functions. This means that if you take something to a certain power and then take the logarithm of the answer, you get back where you started.

The logarithm with base a of a positive number x is defined as:

For x > 0 and 0 < a ≠ 1,
y = loga x if and only if x = ay

In functional notation:

f(x) = loga x is called the logarithmic function with base a

Because the logarithm with base e is used so frequently it has been given a special name Natural logarithm and abbreviation ln. The function is defined as:

f(x) = loge x = ln x   x > 0 is the natural logarithmic function.

## Properties of Logarithms

 1 loga 1 = 0 because a0 = 1 2 loga a = 1 because a1 = a 3 loga ax = x and a log x = x Inverse Properties 4 if loga x = loga y, then x = y One-to-One property

## Properties of Natural Logarithms

All of the properties of logarithms listed above work for Natural Logarithms. There are also some special properties that apply only to the Natural Logarithms.

 1 ln 1 = 0 because e0 = 1 2 ln e = 1 because e1 = e 3 ln ex = x and e ln x = x Inverse Properties 4 if ln x = ln y, then x = y One-to-One property

## Change of Base Formula

Most calculators only have two logarithm keys: log for finding log10 and ln for finding a natural log. To do other problems using a calculator, you need to change the base to either 10 (so you can use the log button) or e (so you can use the ln button).

Let a, b, and x be positive real numbers such that a ≠ 1 and b ≠ 1. Then loga x can be converted to a different base as follows:

 Base b Base 10 Base e loga x = logb x loga x = log10 x loga x = ln x logb a log10 a ln a

## Rules for Working with Logarithms

Let a be a positive number such that a ≠ 1 and let n be a real number. If u and v are positive real numbers, the following rules are true:

Rule

Logarithms

Natural Logarithms

Product

loga (uv) = loga u + loga v ln (uv) = ln u + ln v

Quotient

 loga u = loga u - log a v v
 ln u = ln u - ln v v

Exponent

loga un = loga u + log a v ln un = n ln u Comments, questions, or you just need some help? Send an e-mail to Mrs. Shearer Last Updated: 02/04/2011