 ## Tutoring on Graphing Logarithmic Functions

Learning Objectives:

#### Understand to Graph Logarithmic Functions

Math Tutoring on Graphing Logarithmic Functions

The equations y = logₐ x and x = ay are equivalent. The first equation is in logarithmic form and the second is in exponential form. To sketch the graph of y = logₐ x, we can use the fact that the graphs of inverse functions are the reflections of graph of y = ln (x) each other in the line y = x. The nature of the graph f(x) = logₐ x, a ˃ 1 is that it has one x-intercept and one vertical asymptote. We notice that slowly the graph rises for x ˃ 1. The only difference between this and that of y = log (x) is that this graph increases at a faster rate as x increases. Also, we know that ln (e) = 1 since the base of a natural log function is always e, and e¹= e. Let us look at the graph when a is negative. The graph ranges from -10 to 10.  Negative values for a invert the graph so that y values decrease as x increases. Following are the reason why we use two different logarithmic functions. The common logarithmic function uses 10 as the base of the logarithm. This function is useful for situations like compound interest, the Richter scale, decibel levels, and the exponential growth of a population. The natural logarithmic function uses the irrational constant e (Euler’s constant – 2.71818) as the base of the logarithm.  This function is useful for situations in calculus, statistics (for lines of best fit), and engineering. Thus, each one is useful in distinct situations.

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Hook Questions:

1.    What is the parent function of a logarithmic equation?

2.    How do you graph a natural log?

3.    What is the definition of a logarithmic function?

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