A logarithmic function is the inverse of an exponential function. The general form of an exponential function would be y = b^x, where b is the base of the exponent and x is the exponent or the index. b belongs to positive real numbers and x is any real number. Therefore we see that the domain of an exponential function is all real numbers; whereas the range of this exponential function would be all positive real numbers.
The general form of a logarithmic function would be y = log_b?x, where b is the base of the logarithm. It is the same base that we used above in the exponential function. The x and y have switched places. Therefore, if we were to write the exponential function y = b^x in logarithmic form it would be log_b?y = x.
Graphs of logarithmic functions: To graph logarithmic functions online could be easy, but to graph them manually is not difficult either. Since we know that the logarithmic function is the inverse of an exponential function, we can make a table of values and plot the points, and then join the points with a curve. Let us try to see that with an example.
Example 1: Graph the function y = log_2?x.
Tabulating the results we have:
X (1/4) (1/2) 1 2 4 8
y -2 -1 0 1 2 3
Now we plot those ordered pairs on a graph sheet and run a curve through it. See picture below:
The blue curve above is the graph of the logarithmic function y = log_2?x. The red curve is the graph of the inverse exponential function y = 2^x.
Solving logarithmic functions: There are various methods to solve logarithmic functions. One way is to convert the logarithmic function to the corresponding exponential function and then solve.
Example 2: Solve log_5?125 = y.
Solution: Converting to exponential form we have, 5^y = 125
5^y = 5^3
Since in the above equation the bases are equal, the exponents would also be equal.
Therefore, y = 3 is the answer.
The general form of a logarithmic function would be y = log_b?x, where b is the base of the logarithm. It is the same base that we used above in the exponential function. The x and y have switched places. Therefore, if we were to write the exponential function y = b^x in logarithmic form it would be log_b?y = x.
Graphs of logarithmic functions: To graph logarithmic functions online could be easy, but to graph them manually is not difficult either. Since we know that the logarithmic function is the inverse of an exponential function, we can make a table of values and plot the points, and then join the points with a curve. Let us try to see that with an example.
Example 1: Graph the function y = log_2?x.
Tabulating the results we have:
X (1/4) (1/2) 1 2 4 8
y -2 -1 0 1 2 3
Now we plot those ordered pairs on a graph sheet and run a curve through it. See picture below:
The blue curve above is the graph of the logarithmic function y = log_2?x. The red curve is the graph of the inverse exponential function y = 2^x.
Solving logarithmic functions: There are various methods to solve logarithmic functions. One way is to convert the logarithmic function to the corresponding exponential function and then solve.
Example 2: Solve log_5?125 = y.
Solution: Converting to exponential form we have, 5^y = 125
5^y = 5^3
Since in the above equation the bases are equal, the exponents would also be equal.
Therefore, y = 3 is the answer.
