Understand Logarithms

Confused by the logarithms? Don't worry! A logarithm (log for short) is actually just an exponent in a different form.

logax = y is the same as ay = x.[1]

Steps

  1. Know the difference between logarithmic and exponential equations. This is a very simple first step. If it contains a logarithm (for example: logax = 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: logax = y
    • Exponential: ay = x
  2. Know 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]
  3. Know the difference between a common log and a natural log.
    • Common logs have a base of 10. (for example, log10x). 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.) logex 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, log2x). Hexadecimal logs have the base of 16 (for the example log16x (or log#0fx in the notation of hexadecimal). Logs that have the 64th base are indeed quite complex, and therefore are usually restricted to the Advanced Computer Geometry (ACG) domain.
  4. Know 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.
    • loga(xy) = logax + logay
      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:
      log216 =
      log28*2 =
      log28 + log22
    • loga(x/y) = logax - logay
      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:
      log2(5/3) =
      log25 - log23
    • loga(xr) = r*logax
      If the argument x of the log has an exponent r, the exponent can be moved to the front of the logarithm.

      Example:
      log2(65)
      5*log26
    • loga(1/x) = -logax
      Think about the argument. (1/x) is equal to x-1. Basically this is another version of the previous property.

      Example:
      log2(1/3) = -log23
    • logaa = 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:
      log22 = 1
    • loga1 = 0
      If the argument is one the answer is always Understand the Concept of Zero ( 0 ). This property holds true because any number with an exponent of zero is equal to one.

      Example:
      log31 =0
    • (logbx/logba) = logax
      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 Add Fractions With Like Denominators 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:
      log25 = (log 5/log 2)
  5. Practice 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 = log28 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.

Videos on Properties

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.

Related Articles

Sources and Citations

  1. ↑ Using and Deriving Algebraic Properties of Logarithms
  2. ↑ Logarithms - NDT Resources Center, http://www.ndt-ed.org/EducationResources/Math/Math-Logs.htm
  3. ↑ Logarithms - Wikipedia
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