The equation represents a reflection of the parent function about the x-axis.

When the input is multiplied by the result is a reflection about the y-axis.

When the parent function is multiplied by the result is a reflection about the x-axis.

compresses the parent function vertically by a factor of if.

stretches the parent function vertically by a factor of if.

The equation shifts the parent function vertically.

The equation shifts the parent function horizontally.

The graph of the parent function has an x-intercept at domain range vertical asymptote and.

To find the domain of a logarithmic function, set up an inequality showing the argument greater than zero, and solve for See (Figure) and (Figure).

To find the domain, we set up an inequality and solve for That is, the argument of the logarithmic function must be greater than zero.ДЇor example, consider This function is defined for any values of such that the argument, in this case is greater than zero.

When finding the domain of a logarithmic function, therefore, it is important to remember that the domain consists only of positive real numbers. In Graphs of Exponential Functions we saw that certain transformations can change the range of Similarly, applying transformations to the parent function can change the domain. Just as with other parent functions, we can apply the four types of transformations-shifts, stretches, compressions, and reflections-to the parent function without loss of shape.

Transformations of the parent function behave similarly to those of other functions. In the last section we learned that the logarithmic function is the inverse of the exponential function So, as inverse functions: Recall that the exponential function is defined as for any real number and constant where For instance, what if we wanted to know how many years it would take for our initial investment to double? (Figure) shows this point on the logarithmic graph.ДЇinding the Domain of a Logarithmic FunctionД«efore working with graphs, we will take a look at the domain (the set of input values) for which the logarithmic function is defined. By graphing the model, we can see the output (year) for any input (account balance). We already know that the balance in our account for any year can be found with the equationД«ut what if we wanted to know the year for any balance? We would need to create a corresponding new function by interchanging the input and the output thus we would need to create a logarithmic model for this situation. To illustrate, suppose we invest in an account that offers an annual interest rate of compounded continuously. In other words, logarithms give the cause for an effect. How do logarithmic graphs give us insight into situations? Because every logarithmic function is the inverse function of an exponential function, we can think of every output on a logarithmic graph as the input for the corresponding inverse exponential equation. In Graphs of Exponential Functions, we saw how creating a graphical representation of an exponential model gives us another layer of insight for predicting future events.

Identify the domain of a logarithmic function.

Characteristics of Graphs of Exponential FunctionsД«efore we begin graphing, it is helpful to review the behavior of exponential growth. It gives us another layer of insight for predicting future events. We learn a lot about things by seeing their visual representations, and that is exactly why graphing exponential equations is a powerful tool. Most of the time, however, the equation itself is not enough. Working with an equation that describes a real-world situation gives us a method for making predictions.

Write the equation of an exponential function that has been transformed.ДЄs we discussed in the previous section, exponential functions are used for many real-world applications such as finance, forensics, computer science, and most of the life sciences.

Graph a reflected exponential function.

Graph a stretched or compressed exponential function.

Graph exponential functions shifted horizontally or vertically and write the associated equation.

Sketch a graph of an exponential function.

Determine whether an exponential function and its associated graph represents growth or decay.

Graphmatica logarithmic graphs