When do you use a logarithmic graph

What type s of translation s , if any, affect the range of a logarithmic function? What type s of translation s , if any, affect the domain of a logarithmic function? Shifting the function right or left and reflecting the function about the y-axis will affect its domain. Does the graph of a general logarithmic function have a horizontal asymptote? A horizontal asymptote would suggest a limit on the range, and the range of any logarithmic function in general form is all real numbers.

Domain: Range:. For the following exercises, state the domain and the vertical asymptote of the function. Domain: Vertical asymptote:. For the following exercises, state the domain, vertical asymptote, and end behavior of the function. Domain: ; Vertical asymptote: ; End behavior: as and as. Domain: ; Vertical asymptote: ; End behavior: as , and as ,. For the following exercises, state the domain, range, and x — and y -intercepts, if they exist. If they do not exist, write DNE.

For the following exercises, match each function in Figure with the letter corresponding to its graph. For the following exercises, sketch the graphs of each pair of functions on the same axis. For the following exercises, write a logarithmic equation corresponding to the graph shown. Use as the parent function. Technology For the following exercises, use a graphing calculator to find approximate solutions to each equation.

Let be any positive real number such that What must be equal to? Verify the result. Explore and discuss the graphs of and Make a conjecture based on the result. The graphs of and appear to be the same; Conjecture: for any positive base.

What is the domain of the function Discuss the result. Recall that the argument of a logarithmic function must be positive, so we determine where. From the graph of the function note that the graph lies above the x -axis on the interval and again to the right of the vertical asymptote, that is Therefore, the domain is.

Use properties of exponents to find the x -intercepts of the function algebraically. Show the steps for solving, and then verify the result by graphing the function. Skip to content Exponential and Logarithmic Functions. Learning Objectives In this section, you will: Identify the domain of a logarithmic function. Graph logarithmic functions. Finding the Domain of a Logarithmic Function Before working with graphs, we will take a look at the domain the set of input values for which the logarithmic function is defined.

Recall that the exponential function is defined as for any real number and constant where The domain of is The range of is In the last section we learned that the logarithmic function is the inverse of the exponential function So, as inverse functions: The domain of is the range of The range of is the domain of Transformations of the parent function behave similarly to those of other functions.

Given a logarithmic function, identify the domain. Set up an inequality showing the argument greater than zero. Solve for Write the domain in interval notation. Identifying the Domain of a Logarithmic Shift. Identifying the Domain of a Logarithmic Shift and Reflection. Graphing Logarithmic Functions Now that we have a feel for the set of values for which a logarithmic function is defined, we move on to graphing logarithmic functions.

Figure shows the graph of and Notice that the graphs of and are reflections about the line. Characteristics of the Graph of the Parent Function,. The graphs of three logarithmic functions with different bases, all greater than 1. Draw and label the vertical asymptote, Plot the x- intercept, Plot the key point Draw a smooth curve through the points. State the domain, the range, and the vertical asymptote,. Before graphing, identify the behavior and key points for the graph.

Since is greater than one, we know the function is increasing. The left tail of the graph will approach the vertical asymptote and the right tail will increase slowly without bound. The x -intercept is The key point is on the graph. We draw and label the asymptote, plot and label the points, and draw a smooth curve through the points see Figure. Graphing Transformations of Logarithmic Functions As we mentioned in the beginning of the section, transformations of logarithmic graphs behave similarly to those of other parent functions.

Horizontal Shifts of the Parent Function. Identify the horizontal shift: If shift the graph of left units. If shift the graph of right units. Draw the vertical asymptote Identify three key points from the parent function. Find new coordinates for the shifted functions by subtracting from the coordinate. Label the three points. The Domain is the range is and the vertical asymptote is. The top axis emphasizes the fact the data are logs.

The bottom axis shows the values in the original scale. This labeling follows the advice of William Cleveland with the top and bottom axes interchanged. The data values are spread out better with the logarithmic scale. This is what I mean by responding to skewness of large values. The revenue for Boeing is about 2 6 billion dollars while the revenue for Ford Motor is about 2 7. In Figure 1, the linear scale, the revenue for Ford is the revenue for Boeing plus the difference between these two revenues.

We call this additive. In Figure 2 the difference is multiplicative. This is what I mean by saying that we use logarithmic scales to show multiplicative factors. The previous example showed both responding to large values and multiplicative factors. The next example just describes rates of change.

Suppose we had one widget in and doubled the number each year. The following charts show the number of widgets on a linear and logarithmic scale:. The linear scale shows the absolute number of widgets over time while the logarithmic scale shows the rate of change of the number of widgets over time.

The bottom chart of Figure 4 makes it much clearer that the rate of change or growth rate is constant. Between each major value on the logarithmic scale, the hashmarks become increasingly closer together with increasing value.

The advantages of using a logarithmic scale are twofold. Firstly, doing so allows one to plot a very large range of data without losing the shape of the graph. Secondly, it allows one to interpolate at any point on the plot, regardless of the range of the graph. Similar data plotted on a linear scale is less clear.

A key point about using logarithmic graphs to solve problems is that they expand scales to the point at which large ranges of data make more sense. We now rely on the properties of logarithms to re-write the equation. Privacy Policy. Skip to main content. Exponents, Logarithms, and Inverse Functions. Search for:.

Graphs of Exponential and Logarithmic Functions. Learning Objectives Describe the properties of graphs of exponential functions. This is known as exponential growth. This is known as exponential decay. Key Terms exponential growth : The growth in the value of a quantity, in which the rate of growth is proportional to the instantaneous value of the quantity; for example, when the value has doubled, the rate of increase will also have doubled.

Logarithms are everywhere. Ever use the following phrases? Mama mia! Ok, ok, we get it: what are logarithms about? Logarithms find the cause for an effect, i. By the way, the notion of "cause and effect" is nuanced.

Why is bigger than ? Logarithms put numbers on a human-friendly scale. Logarithms count multiplication as steps Logarithms describe changes in terms of multiplication: in the examples above, each step is 10x bigger.

Show me the math Time for the meat: let's see where logarithms show up! Six-figure salary or 2-digit expense We're describing numbers in terms of their digits, i. Try it out here: Order of magnitude We geeks love this phrase. Interest Rates How do we figure out growth rates? My two favorite interpretations of the natural logarithm ln x , i. Google conveys a lot of information with a very rough scale Measurement Scale: Richter, Decibel, etc.

Logarithmic Graphs You'll often see items plotted on a "log scale". Onward and upward If a concept is well-known but not well-loved, it means we need to build our intuition. In my head: Logarithms find the root cause for an effect see growth, find interest rate They help count multiplications or digits, with the bonus of partial counts k is a 6. Continuous Growth What does an exponent really mean?

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