Which of the following is a logarithmic function?
$y=\log _3 x$
$y=3^x$
$y=x+3$
$y=x^3$
Answer (1)
A logarithmic function has the form y = lo g b x , where b is a positive constant not equal to 1.
Compare the given functions to the definition of a logarithmic function.
Identify y = lo g 3 x as the logarithmic function.
The final answer is y = lo g 3 x .
Explanation
Understanding the Problem We are given four functions: y = lo g 3 x , y = 3 x , y = x + 3 , and y = x 3 . We need to identify which one is a logarithmic function.
Definition of Logarithmic Function A logarithmic function is of the form y = lo g b x , where b is a positive constant not equal to 1.
Identifying the Logarithmic Function Comparing the given functions with the definition of a logarithmic function:
y = lo g 3 x matches the definition of a logarithmic function with base 3.
y = 3 x is an exponential function.
y = x + 3 is a linear function.
y = x 3 is a power function.
Conclusion Therefore, the logarithmic function among the given options is y = lo g 3 x .
Examples
Logarithmic functions are used in many real-world applications, such as measuring the intensity of earthquakes (Richter scale), the loudness of sounds (decibels), and the acidity of solutions (pH scale). Understanding logarithmic functions helps us to work with quantities that vary over a very large range of values. For example, the Richter scale uses logarithms to compress the range of earthquake magnitudes, making it easier to compare the relative sizes of different earthquakes. A magnitude 6 earthquake is ten times stronger than a magnitude 5 earthquake.
