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 :
A only partially related value is the discrete logarithm, used in cryptography via modular arithmetic. It's the lowest value such that, for given being integers (as well as the unknowns being integer).
Its related to the usual logarithm, by the fact that if isn't an integer power of then is a lower bound on
- 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