# Convert Logarithms and Exponentials

 The logarithmic functions logb x and the exponential functions bx are inverse of each other, hence y = logb x is equivalent to x = by where b is the common base of the exponential and the logarithm. The above equivalence helps in solving logarithmic and exponential functions and needs a deep understanding. Examples, of how the above relationship between the logarithm and exponential may be used to transform expressions, are presented below. Example 1 : Change each logarithmic expression to an exponential expression. 1. log3 27 = 3 2. log36 6 = 1 / 2 3. log2 (1 / 8) = -3 4. log8 2 = 1 / 3 Solution to Example 1: 1.     The logarithmic form log3 27 = 3 is equivalent to the exponential form 27 = 33 2.     The logarithmic form log36 6 = 1 / 2 is equivalent to the exponential form 6 = 361/2 3.     log2 (1 / 8) = -3 in exponential form is given by 1 / 8 = 2-3 4.     log8 2 = 1 / 3 in exponential form is given by 2 = 81/3 Example 2 : Change each exponential expression to logarithmic expression. 1. 34 = 81 2. 41/2 = 2 3. 3-3 = 1 / 27 4. 103 = 1000 Solution to Example 2: 1.     The exponential form 34 = 81 is equivalent to the logarithmic form 4 = log3 81 2.     The exponential form 41/2 = 2 is equivalent to the logarithmic form 1 / 2 = log4 2 3.     3-3 = 1 / 27 in logarithmic form is given by -3 = log3 (1/ 27) 4.     103 = 1000 in logarithmic form is given by 3 = log10 1000 Example 3 : Solve for x the following equations. 1. log3 x = 5 2. log2 (x - 3) = 2 3. 2 log3 (- x + 1) = 6 Solution to Example 3: 1.     To solve the equation log3 x = 5, rewrite it into exponential form x = 35 2.     Rewrite the equation log2 (x - 3) = 2 into exponential form x - 3 = 22 = 4 Solve for x x = 4 + 3 = 7 3.     Divide all terms of the equation 2 log3 (- x + 1) = 6 by 2 log3 (- x + 1) = 3 Rewrite the equation obtained in exponential form - x + 1 = 33 = 27 Solve for x x = - 26 More references and links related to the logarithmic functions. Solve Exponential and Logarithmic Equations (self test).