Logarithms and exponents are very closely related. In fact, they are inverse functions. This means that logarithms can be used to reverse the result of exponentiation and vice versa, just as addition can be used to reverse the result of subtraction. Thus, if we have , then taking the logarithm with base on both sides will give us .
We would read this as "the logarithm of b, base a, is x". For example, we know that . To express the same fact in logarithmic notation we would write .
Depending on the field, the symbol without a base can have different meanings. Sometimes in high schools, the symbol is used to refer to a base 10 logarithm. Thus, can mean . In these contexts, the symbol (an abbreviation of the French "logarithme normal," meaning "natural logarithm") is introduced to refer to the logarithm base e, or natural logarithm. However, the choice of base 10 is arbitrary, and convenient only for computations in a base-10 number system. The natural logarithm, however, has many convenient mathematical properties, so practicing mathematicians often take the symbol to mean the natural logarithm and do not use the symbol . (This is an example of conflicting mathematical conventions.) In addition, the notation is often used by combinatorists and computer scientists to refer to the logarithm base . Occasionally, the base of the logarithms is irrelevant.
We can use the properties of exponents to build a set of properties for logarithms.
We know that . We let and . This also makes . From , we have , and from , we have . So, . But we also have from that . Thus, we have found two expressions for establishing the identity:
Using the laws of exponents, one can derive and prove the following identities:
- (the change of base formula)
These formulas also have a number of common special cases:
- (sometimes known as the inverse property of logarithms)
Powerful use of logarithms
Some of the real powerful uses of logarithms, come down to never having to deal with massive numbers. ex. : would be a pain to have to calculate any time you wanted to use it (say in a comparison of large numbers). its natural logarithm though (partly due to left to right parenthesized exponentiation) is only 7 digits before the decimal point. Comparing the logs of the numbers to a given precision can allow easier comparision than computing and comparing the numbers themselves. Logs also allow (with repetition) to turn left to right exponentiation into power towers (especially useful for tetration (exponentiation repetition with the same exponent)). ex.
Therefore by : and identities 1 and 2 above ( 2 being used twice) we get:
such that :
- Evaluate .
- Evaluate .
- Simplify where .
The natural logarithm is the logarithm with base e. It is usually denoted , an abbreviation of the French logarithme normal, so that However, in higher mathematics such as complex analysis, the base 10 logarithm is typically disposed with entirely, the symbol is taken to mean the logarithm base e and the symbol is not used at all. (This is an example of conflicting mathematical conventions.)
can also be defined as the area under the curve between 1 and a, or .
All logarithms are undefined in nonpositive reals, as they are complex. From the identity , we have . Additionally, for positive real .
- What is the value of for which ?
- Positive integers and satisfy the condition Find the sum of all possible values of .
- The sequence is geometric with and common ratio where and are positive integers. Given that find the number of possible ordered pairs
Two-minute Intro to Logarithms