Convert Logarithms and Exponentials
Relationship Between Exponential and LogarithmThe logarithmic functionslogb x and the exponential functionsbx are inverse of each other, hencewhere 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 1Change 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:
2. The logarithmic form log36 6 = 1 / 2 is equivalent to the exponential form 3. log2 (1 / 8) = -3 in exponential form is given by 4. log8 2 = 1 / 3 in exponential form is given by
Example 2Change 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:
2. The exponential form 41/2 = 2 is equivalent to the logarithmic form 3. 3-3 = 1 / 27 in logarithmic form is given by 4. 103 = 1000 in logarithmic form is given by
Example 3Solve for x the following equations.1. log3 x = 5 2. log2 (x - 3) = 2 3. 2 log3 (- x + 1) = 6
Solution to Example 3:
2. Rewrite the equation log2 (x - 3) = 2 into exponential form x = 4 + 3 = 7 3. Divide all terms of the equation 2 log3 (- x + 1) = 6 by 2 Rewrite the equation obtained in exponential form - x + 1 = 33 = 27 Solve for x x = - 26 More References and Links Related to the Logarithmic FunctionsSolve Exponential and Logarithmic Equations (self test).Rules of Logarithms and Exponentials - Questions with Solutions. |
