Graphing More Complex Logarithmic Functions
Description/Explanation/Highlights
Video Description
This video explains how to graph more complex Log functions by isolating the log, solving and using the inverse.
Steps and Key Points to Remember
To graph more complex logarithmic functions using the inverse, follow these steps:
- Graphing any logarithmic function is similar to graphing the parent function but may involve more steps to isolate the log in the beginning and steps to isolate a variable after rewriting into exponential form.
- To graph a logarithmic function like \(y=2+\log_3(x-1)\), first isolate the log, base and argument.
- In this example, subtract 2 from both sides to get rid of the extra term on the side with the Log i.e. \(y-2=\log_3(x-1)\)
- Now rewrite the function into exponential form. i.e. \(3^{y-2}=x-1\)
- Since log and exponential are inverses, we usually switch x and y and solve for y at the point but it is usually easier after rewriting in exponential form to simply pick the y-values first and plug them in to find x instead of switching x and y and then switching back.
- Before selecting y-values and solving for x, we should first solve the exponential equation for x. In this example, simply add 1 to both sides of the equation i.e. \(3^{y-2}+1=x\)
- Now create a table of values by selecting some y-values and finding their corresponding x-values.
- Graph these point and connect them with a smooth curve and you will have graphed the original log function.
- When a number is added or subtracted inside the log function, the graph will be shifted right (-) or left (+) from the original parent. The new vertical asymptote will be shifted this many places from the y-axis. It is often helpful to draw the asymptote in as a dotted line for reference when graphing.
- Remember, it is usually easier after rewriting in exponent form to simply pick the y-values first and plug them in to find x instead of switch x and y and then switching back. the result is the same.
Here are some key points to keep in mind when using the inverse to graph logarithmic functions.
- Log functions and exponential functions are inverses of each other.
- Finding inverses of graphs is as simple as switching the x- and y-values and graphing the new coordinates.
- Rewriting from log to exponential and switching x and y finds the inverse of the log.
- Switching the x- and y-values in a table of the exponential form takes us back to a log.
- Finding the inverse and switching x- and y-values back works to graph a log of any base.
- Since the vertical asymptote of a logarithmic parent is at x = 0 (the y-axis), it may be helpful to locate the new vertical asymptote in more complex functions.
Video Highlights
- 00:00 Single example of graphing a complex logarithmic function.
- To watch this video on YouTube in a new window with clickable highlights, click here
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