The
logarithmic functions log_{b} x and the exponential functions b^{x} are inverse of each other, hence
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 1 : Change 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
Solution to Example 1:
1. The logarithmic form log_{3} 27 = 3 is equivalent to the exponential form
27 = 3^{3}
2. The logarithmic form log_{36} 6 = 1 / 2 is equivalent to the exponential form
6 = 36^{1/2}
3. log_{2} (1 / 8) = -3 in exponential form is given by
1 / 8 = 2^{-3}
4. log_{8} 2 = 1 / 3 in exponential form is given by
2 = 8^{1/3}
Example 2 : Change each exponential expression to logarithmic expression.
1. 3^{4} = 81
2. 4^{1/2} = 2
3. 3^{-3} = 1 / 27
4. 10^{3} = 1000
Solution to Example 2:
1. The exponential form 3^{4} = 81 is equivalent to the logarithmic form
4 = log_{3} 81
2. The exponential form 4^{1/2} = 2 is equivalent to the logarithmic form
1 / 2 = log_{4} 2
3. 3^{-3} = 1 / 27 in logarithmic form is given by
-3 = log_{3} (1/ 27)
4. 10^{3} = 1000 in logarithmic form is given by
3 = log_{10} 1000
Example 3 : Solve for x the following equations.
1. log_{3} x = 5
2. log_{2} (x - 3) = 2
3. 2 log_{3} (- x + 1) = 6
Solution to Example 3:
1. To solve the equation log_{3} x = 5, rewrite it into exponential form
x = 3^{5}
2. Rewrite the equation log_{2} (x - 3) = 2 into exponential form
x - 3 = 2^{2} = 4
Solve for x
x = 4 + 3 = 7
3. Divide all terms of the equation 2 log_{3} (- x + 1) = 6 by 2
log_{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 functions.
Solve Exponential and Logarithmic Equations (self test).