Understanding logarithms fast (5 min) by mathOgenius

How to Understand Logarithms


  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.
  3. Know the difference between a common log and a natural log.
    • Common logshave 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)nas 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 theEmathematical base constant.Binarylogs have a base of 2 (for the example, log2x).Hexadecimallogs have the base of 16. Logs that have the 64thbase are used in 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 baseaand the argument are positive. Also the baseacannot 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,xandy, 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.)
      log216 =
      log28*2 =
      log28 + log22
    • loga(x/y) = logax - logay
      A log of two numbers being divided by each other,xandy, can be split into two logs: the log of the dividendxminus the log of the divisory.
      log2(5/3) =
      log25 - log23
    • loga(xr) = r*logax
      If the argumentxof the log has an exponentr, the exponent can be moved to the front of the logarithm.
    • loga(1/x) = -logax
      Think about the argument. (1/x) is equal to x-1. Basically this is another version of the previous property.
      log2(1/3) = -log23
    • logaa = 1
      If the baseaequals the argumentathe answer is 1. This is very easy to remember if one thinks about the logarithm in exponential form. How many times should one multiplyaby itself to geta? Once.
      log22 = 1
    • loga1 = 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.
      log31 =0
    • (logbx/logba) = logax
      This is known as "Change of Base".One log divided by another, both with the same baseb, is equal to a single log. The argumentaof the denominator becomes the new base, and the argumentxof 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.
      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.

Community Q&A

  • Question
    Can you provide some exam style questions of logarithms?
    wikiHow Contributor
    Community Answer
    Here are some few practice problems : 1. log(3 + 2 log(1+x) ) =0 , Find x 2. If (2.5)^x = 0.025)^y, find x and y.
  • Question
    How do I find an answer for 6=log(x/5)?
    wikiHow Contributor
    Community Answer
    You must expand the expression to 6=log(x)-log(5). This is an example of the quotient property of a logarithm log(a/b)=log(a)-log(b). You then do log(5), which is approximately 0.699, so 6=log(x)-0.699. Add 0.699 to both sides to get 6.699=log(x). Then rewrite it in exponential form as 10^6.699=x and do the rest.
  • Question
    How do you get ln(y)?
    wikiHow Contributor
    Community Answer
    The ln(y) function is similar to a log function. A log function uses a base of ten (log base ten of x is often written log(x)), unless otherwise specified. A function ln(x) is just a logarithm with a base of e, a number that is similar to pi in the fact that it is a mathematical constant. The letter e represents the number 2.71828. So, ln(y) is equivalent to log of y with a base of e, or log base e of y.
  • Question
    Where do I place X in a logarithm?
    wikiHow Contributor
    Community Answer
    X is a variable. In a logarithm, the value to be found is denoted by X. (It can be either the base or argument.)
  • Question
    Can I calculate logs of negative numbers?
    wikiHow Contributor
    Community Answer
    You can find the solution of a negative log, however the number that you find will not be rational so for all intents and purposes you cannot.
  • Question
    What is the purpose of logarithms?
    Sage Vetter
    Community Answer
    Logarithms provide a tool to solve problems. Another way to think of this, logarithms rewrite problems to indicate exponential numbers without using any powers in the actual equation. Logarithms provides greater access to all the numbers in the equation. This is the basic purpose of a log.
  • Question
    What is the value of x in the equation log (x+1)+log (x-1)=log 3?
    wikiHow Contributor
    Community Answer
    Simplify the left side to a single log (x^2-1). Then take the antilog of both sides (x^2-1=3). Solve resultant polynomial equation (x=2 or x= -2). Check solutions against original equation for domain issues. For example, if x=-2, log (x-1) is undefined, so it is not the correct solution.
Unanswered Questions
  • What is a good technique to use for solving logs using different bases?
  • What is the solution to 3 log4?
  • What is the answer for (Log 9 base 2)*(Log 7 base 3)*(Log 8 base 7)?
  • How do I show that: logab=1/logba?
  • How do you convert something in exponential form to logarithmic form? A^5=686, for example.
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Videos on Properties

  • "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 wikiHows

Sources and Citations

  1. Using and Deriving Algebraic Properties of Logarithms
  2. Logarithms - NDT Resources Center,

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Date: 15.12.2018, 06:35 / Views: 43333