How Are Logs Solved?


Quick Answer

Mathematical logs, or logarithmic equations, are easiest to solve by rewriting or stating the equations as exponents and then solving for the variable. This is because logarithms are the mathematical opposite of exponents or exponential equations.

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Full Answer

Just as subtraction is the mathematical "opposite" of addition, logs are the "opposite" of exponents. In other words, they undo or inverse an exponential equation. For this reason, many people find it easy to think of logs in terms of their relationship to exponents. There are two basic ways in which to translate logs into exponents, by reading the logs as exponents and by using a formula to convert them into exponential equations

Reading logs as exponents

  1. Understand the elements of the log
  2. Think of the elements of the basic logarithmic equation, logb(y) = x, in the following terms: b is the base, y is the augment, x is the "equals."
  3. Convert the equation to an exponent
  4. Convert the three terms as follows: the base remains the base, the augment becomes the "equals," the "equals" becomes the exponent or "power."
  5. Read the equation as an exponent
  6. Therefore, the logarithm log4(64) = x becomes, "What power (exponent), when put to 4 equals 64?"
  7. Solve the exponential equation
  8. 4 to the third power equals 64. Therefore, the answer is 3.

Converting logarithmic equations to exponential equations

  1. Convert the log to an exponent
  2. Use the formulaic relationship between a log and an exponent to convert the logarithmic equation to an exponential equation. Therefore, because logb(y) = x is equivalent to y = bx, log4(64) = x is equivalent to 64 = 4x.
  3. Solve the exponential equation
  4. 4x4x4 = 64. Therefore, the answer is 3.
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