How to Derive Logarithms

Logarithms are a principal component of higher level math. The logarithm of any number identifies another value that is related to, or a function of, the original value. Logarithms are of the general form log a (b) = x, where a is the base, x is the power that the base is raised to, and b is the value for which the logarithm is being calculated. Logarithms, therefore, can be derived from exponential expressions using these properties.

Instructions

    • 1

      Identify an exponential expression for which a logarithm is desired. For example, assume the logarithmic form of the expression 2^3 = x.

    • 2

      Substitute the values into the general logarithmic form. In the exponential equation example, the number 2 is identified as the base, while the exponent would be the value that the base is raised to. Using the properties of the general logarithm expression identified earlier, the derived logarithm of the exponential expression would be:

      [2^x = 3] = [log 3 (2) = x]

    • 3

      Understand the significance of logarithmic form. The logarithm of a number essentially identifies the power that a specific number, or base, must be raised to produce a resulting number. It is important to realize that logarithmic form and exponential form are interchangeable, as this property is frequently used in mathematics to solve for unknown variables. For example, the expression log 2 (x) = 3 can be converted to 2^3 = x to solve for the unknown x variable.

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