How to Find a Log With Variables in an Exponent

Exponents in math denote how many times a number, called the base, should be multiplied by itself. For example, 4^2 equals 4 * 4 and x^3 equals x * x * x. When the base is known but the exponent is a variable, it is called an exponential equation. An exponential equation is set equal to a constant (number). If that constant can be converted to exponential form, the two exponents can simply be set equal to each other. For example, 2^x = 16 becomes 2^x = 4^2 and then x = 2. If the constant can't be converted, logarithms become necessary for solving.

Instructions

1. • 1

     Solve an exponential equation of the form b^x = a (where "b" is the base, "x" is the variable exponent and "a" is the constant) by converting it to the logarithmic form of x * ln(b) = ln(a), where "ln" equals natural log. Solve the equation for "x".

   • 2

     Solve the exponential equation 2^x = 55. Convert to logarithmic form x * ln(2) = ln(55). Divide ln(2) from both sides to isolate the variable: x = ln(55) / ln(2).

   • 3

     Use a calculator to carefully input the division problem and solve "x": x = 4.00733319 / 0.693147181 = 5.78 (rounded).