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.
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 = log_{a} x if and only if x = a^{y}
In functional notation:
f(x) = log_{a} 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) = log_{e} x = ln x x > 0^{ }is the natural logarithmic function.
1.  log_{a} 1 = 0  because a^{0} = 1 
2.  log_{a} a = 1  because a^{1} = a 
3.  log_{a} a^{x} = x and a ^{log x} = x  Inverse Properties 
4.  if log_{a} x = log_{a} y, then x = y  OnetoOne property 
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 e^{0} = 1 
2.  ln e = 1  because e^{1} = e 
3.  ln e^{x} = x and e ^{ln x} = x  Inverse Properties 
4.  if ln x = ln y, then x = y  OnetoOne property 
Most calculators only have two logarithm keys: log for finding log_{10} 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 log_{a} x can be converted to a different base as follows:
Base b 
Base 10 
Base e 

log_{a} x = 
log_{b} x 
log_{a} x = 
log_{10} x 
log_{a} x = 
ln x 

log_{b} a  log_{10} a  ln a 
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 
log_{a} (uv) = log_{a} u + log_{a} v  ln (uv) = ln u + ln v  
Quotient 



Exponent 
log_{a} u^{n} = log_{a} u + log_{ a} v  ln u^{n }= n ln u 
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