Domain And Range Of Exponential And Logarithmic Functions

3 x 0 so the range would be y 4.

Domain and range of exponential and logarithmic functions. Consider what the inverse of the. The inverse of the exponential function y a x is x a y the logarithmic function y log a x is defined to be equivalent to the exponential equation x a y. The reason a 0 is that if it is negative the function is undefined for 1 x 1 restricting a to positive values allows the function to have a domain of all real numbers. Y log a x only under the following conditions.

When evaluating a logarithmic function with a calculator you may have noticed that the only options are log 10 or log called the common logarithm or ln which is the natural logarithm. However its range is such that y r. Logarithmic functions are the inverses of exponential functions. X 5 is your domain.

Exponential functions have the general form y f x a x where a 0 a 1 and x is any real number. A the domain is all real numbers since 3 x is defined for all real x. In general the function y log b x where b x 0 and b 1 is a continuous and one to one function. Recall that the exponential function is defined as latex y b x latex for any real number x and constant latex b 0 latex latex b ne 1 latex where.

An exponential function is a function in which the independent variable is an exponent. However exponential functions and logarithm functions can be expressed in terms of any desired base b. The range of log x is all real numbers so the range in this case is the real numbers. The points 0 1 and 1 a always lie on the exponential function s graph while 1 0 and b 1 always lie on the logarithmic function s graph.

Before working with graphs we will take a look at the domain the set of input values for which the logarithmic function is defined. B to take log x x must be positive so x 5 0. X a y a 0 and a 1 it is called the logarithmic function with base a. The domain of y is latex left infty infty right latex.

The range and the domain of the two functions are exchanged. Remember that since the logarithmic function is the inverse of the exponential function the domain of logarithmic function is the range of exponential function and vice versa. Here we may think that if the base is not 10 what could be the range of the logarithmic functions. Find the domain and range for f x in x 5 solution.

Product and quotient rules of the exponential and the logarithm functions follow from each other.