How to Understand Logarithms
Confused by the logarithms? Don't worry! A logarithm (log for short) is actually just an exponent in a different form.
log_{a}x = y is the same as a^{y} = x.^{[1]}
Steps
- 1Know the difference between logarithmic and exponential equations. This is a very simple first step. If it contains a logarithm (for example: log_{a}x = y) it is logarithmic problem. A logarithm is denoted by the letters "log". If the equation contains an exponent (that is, a variable raised to a power) it is an exponential equation. An exponent is a superscript number placed after a number.
- Logarithmic: log_{a}x = y
- Exponential: a^{y} = x
- 2Know the parts of a logarithm. The base is the subscript number found after the letters "log"--2 in this example. The argument or number is the number following the subscript number--8 in this example. Lastly, the answer is the number that the logarithmic expression is set equal to--3 in this equation.^{[2]}
- 3Know the difference between a common log and a natural log.
- Common logs have a base of 10. (for example, log_{10}x). If a log is written without a base (as log x), then it is assumed to have a base of 10.
- Natural logs: These are logs with a base of e. e is a mathematical constant that is equal to the limit of (1 + 1/n)^{n} as n approaches infinity, approximately 2.718281828. (It has many more digits than those written here.) log_{e}x is often written as ln x.
- Other Logs: Other logs have the base other than that of the common log and the E mathematical base constant. Binary logs have a base of 2 (for the example, log_{2}x). Hexadecimal logs have the base of 16 (for the example log_{16}x (or log_{#0f}x in the notation of hexadecimal). Logs that have the 64^{th} base are indeed quite complex, and therefore are usually restricted to the Advanced Computer Geometry (ACG) domain.
- 4Know and apply the properties of logarithms. The properties of logarithms allow you to solve logarithmic and exponential equations that would be otherwise impossible. These only work if the base a and the argument are positive. Also the base a cannot be 1 or 0. The properties of logarithms are listed below with a separate example for each one with numbers instead of variables. These properties are for use when solving equations.
- log_{a}(xy) = log_{a}x + log_{a}y
A log of two numbers, x and y, that are being multiplied by each other can be split into two separate logs: a log of each of the factors being added together. (This also works in reverse.)
Example:
log_{2}16 =
log_{2}8*2 =
log_{2}8 + log_{2}2 - log_{a}(x/y) = log_{a}x - log_{a}y
A log of a two numbers being divided by each other, x and y, can be split into two logs: the log of the dividend x minus the log of the divisor y.
Example:
log_{2}(5/3) =
log_{2}5 - log_{2}3 - log_{a}(x^{r}) = r*log_{a}x
If the argument x of the log has an exponent r, the exponent can be moved to the front of the logarithm.
Example:
log_{2}(6^{5})
5*log_{2}6 - log_{a}(1/x) = -log_{a}x
Think about the argument. (1/x) is equal to x^{-1}. Basically this is another version of the previous property.
Example:
log_{2}(1/3) = -log_{2}3 - log_{a}a = 1
If the base a equals the argument a the answer is 1. This is very easy to remember if one thinks about the logarithm in exponential form. How many times should one multiply a by itself to get a? Once.
Example:
log_{2}2 = 1 - log_{a}1 = 0
If the argument is one the answer is always zero. This property holds true because any number with an exponent of zero is equal to one.
Example:
log_{3}1 =0 - (log_{b}x/log_{b}a) = log_{a}x
This is known as "Change of Base".^{[3]} One log divided by another, both with the same base b, is equal to a single log. The argument a of the denominator becomes the new base, and the argument x of the numerator becomes the new argument. This is easy to remember if you think about the base as the bottom of an object and the denominator as the bottom of a fraction.
Example:
log_{2}5 = (log 5/log 2)
- log_{a}(xy) = log_{a}x + log_{a}y
- 5Practice using the properties. These properties are best memorized by repeated use when solving equations. Here's an example of an equation that is best solved with one of the properties:
4x*log2 = log8 Divide both sides by log2.
4x = (log8/log2) Use Change of Base.
4x = log_{2}8 Compute the value of the log.
4x = 3 Divide both sides by 4. x = 3/4 Solved. This is very helpful. I now understand logs.
Tips
- "2.7jacksonjackson" is a useful mnemonic device for e. 1828 is the year Andrew Jackson was elected, so the mnemonic stands for 2.718281828.
Sources and Citations
- ↑ Using and Deriving Algebraic Properties of Logarithms
- ↑ Logarithms - NDT Resources Center, http://www.ndt-ed.org/EducationResources/Math/Math-Logs.htm
- ↑ Logarithms - Wikipedia