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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. In this section we will discuss the values for which a logarithmic function is defined, and then turn our attention to graphing the family of logarithmic functions.
Before working with graphs, we will take a look at the domain (the set of input values) for which the logarithmic function is defined.
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. We draw and label the asymptote, plot and label the points, and draw a smooth curve through the points (see [link]). As we mentioned in the beginning of the section, transformations of logarithmic graphs behave similarly to those of other parent functions. In the above problem there was a reflection about the x-axis as well as a shift to the left 3 units. This is an algebra study guide and problem solver designed to supplement your algebra 1 textbook. This is a€?Logarithmic Functions and Their Graphsa€?, section 7.3 from the book Advanced Algebra (v. This content was accessible as of December 29, 2012, and it was downloaded then by Andy Schmitz in an effort to preserve the availability of this book.
PDF copies of this book were generated using Prince, a great tool for making PDFs out of HTML and CSS. For more information on the source of this book, or why it is available for free, please see the project's home page. DonorsChoose.org helps people like you help teachers fund their classroom projects, from art supplies to books to calculators. We begin with the exponential function defined by f(x)=2x and note that it passes the horizontal line test. To find the inverse algebraically, begin by interchanging x and y and then try to solve for y. This gives us another transcendental function defined by fa?’1(x)=log2a€‰x, which is the inverse of the exponential function defined by f(x)=2x.
The domain consists of all positive real numbers (0,a?z) and the range consists of all real numbers (a?’a?z,a?z).


In general, given base b>0 where ba‰ 1, the logarithm base bThe exponent to which the base b is raised in order to obtain a specific value. It is useful to note that the logarithm is actually the exponent y to which the base b is raised to obtain the argument x. If the base of a logarithm is e, the logarithm is called the natural logarithmThe logarithm base e, denoted lna€‰x.. The domain consists of positive real numbers, (0,a?z) and the range consists of all real numbers, (a?’a?z,a?z). Note: Finding the intercepts of the graph in the previous example is left for a later section in this chapter. Next, consider exponential functions with fractional bases, such as the function defined by f(x)=(12)x. In this case the shift left 3 units moved the vertical asymptote to x=a?’3 which defines the lower bound of the domain. The base-b logarithmic function is defined to be the inverse of the base-b exponential function.
Logarithmic functions with definitions of the form f(x)=logba€‰x have a domain consisting of positive real numbers (0,a?z) and a range consisting of all real numbers (a?’a?z,a?z). To graph logarithmic functions we can plot points or identify the basic function and use the transformations.
In this section we will solve typical word problems that involve exponential growth or decay. If k is positive then we will have a growth model and if k is negative then we will have a decay model.
The basic idea is to first determine the given information then substitute the appropriate values into the formula and evaluate. The key step in this process is to apply the common logarithm to both sides so that we can apply the power rule and solve for time t. Example: How long will it take $30,000 to accumulate to $110,000 in a trust that earns a 10% return compounded semiannually? The key step in this process is to apply the natural logarithm to both sides so that we can apply the power rule and solve for t. Example: How long will it take $30,000 to accumulate to $110,000 in a trust that earns a 10% annual return compounded continuously?


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. 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. When finding the domain of a logarithmic function, therefore, it is important to remember that the domain consists only of positive real numbers.
First, we move the graph left 2 units, then stretch the function vertically by a factor of 5, as in [link]. See the license for more details, but that basically means you can share this book as long as you credit the author (but see below), don't make money from it, and do make it available to everyone else under the same terms. However, the publisher has asked for the customary Creative Commons attribution to the original publisher, authors, title, and book URI to be removed. If the base is 10, the logarithm is called the common logarithmThe logarithm base 10, denoted loga€‰x..
Assuming exponential growth, how long will it take the population to grow from 40 specimens to 500?
If it is determined that an old bone contains 85% of it original carbon-14 how old is the bone? In the previous problem, notice that the principal was not given and that the variable P cancelled. 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.
You may also download a PDF copy of this book (41 MB) or just this chapter (5 MB), suitable for printing or most e-readers, or a .zip file containing this book's HTML files (for use in a web browser offline). Recall that if (x,y) is a point on the graph of a function, then (y,x) will be a point on the graph of its inverse.
In addition, f(b)=logba€‰b=1 and so (b,1) is a point on the graph no matter what the base is.
For instance, what if we wanted to know how many years it would take for our initial investment to double?



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