# Convert Logarithms and Exponentials

## Relationship Between Exponential and LogarithmThe logarithmic functionslog and the exponential functions_{b} xb are inverse of each other, hence
^{x} y = log_{b} x is equivalent to x = b^{y} 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 1Change each logarithmic expression to an exponential expression.1. log _{3} 27 = 3
2. log _{36} 6 = 1 / 2
3. log _{2} (1 / 8) = -3
4. log _{8} 2 = 1 / 3
^{3}
2. The logarithmic form log _{36} 6 = 1 / 2 is equivalent to the exponential form
^{1/2}
3. log _{2} (1 / 8) = -3 in exponential form is given by
^{-3}
4. log _{8} 2 = 1 / 3 in exponential form is given by
^{1/3}
## Example 2Change each exponential expression to logarithmic expression.1. 3 ^{4} = 81
2. 4 ^{1/2} = 2
3. 3 ^{-3} = 1 / 27
4. 10 ^{3} = 1000
_{3} 81
2. The exponential form 4 ^{1/2} = 2 is equivalent to the logarithmic form
_{4} 2
3. 3 ^{-3} = 1 / 27 in logarithmic form is given by
_{3} (1/ 27)
4. 10 ^{3} = 1000 in logarithmic form is given by
_{10} 1000
## Example 3Solve for x the following equations.1. log _{3} x = 5
2. log _{2} (x - 3) = 2
3. 2 log _{3} (- x + 1) = 6
^{5}
2. Rewrite the equation log _{2} (x - 3) = 2 into exponential form^{2} = 4
x = 4 + 3 = 7 3. Divide all terms of the equation 2 log _{3} (- x + 1) = 6 by 2_{3} (- x + 1) = 3
Rewrite the equation obtained in exponential form - x + 1 = 3 ^{3} = 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. |