__Math Tutoring on Graphing Logarithmic Functions__

The equations **y = logₐ x and x = a**^{y} 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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