04Bloggraphs

toc = Logarithmic Graphs =

The graph of y = log a x (where a is positive but excluding 1) is called a ** logarithmic graph **.

Logarithmic graphs (base 10) are used in measuring levels of noise (decibels) and strength of earthquakes (Richter Scale)

y = log a x
 * has a vertical asymptote at x = 0
 * is only defined for x > 0
 * has an x-intercept at (1, 0)
 * goes through the point (a, 1)

Graph of Log Base 2
The graph of y = log 2 x is shown here:

Asymptote: x = 0

x-intercept: (1, 0)

A second point is at (2, 1) because log 2 2 = 1

No y-intercepts

No turning points

Domain: x Î R +

Range: y Î R

This is a strictly increasing graph (gradient is always positive).



Comparing Log and Exponential Graphs
We have previously discussed how y = a x and y = log a x are reverse operations.

The graphs of y = a x and y = log a x are the **inverse** of each other.

This means that they are a reflection across the line y = x. OR the x and y coordinates of the individual points are swapped so (x, y) → (y, x)

Log Graphs with a > 1


All logarithmic graphs, y = log a x, where a > 1 have the following in common:
 * Asymptote at x = 0: as x → 0, y → – ¥
 * x-intercept at (1, 0): log a 1 = 0
 * 2nd point at (a, 1): log a a = 1
 * Strictly Increasing graph
 * As x → + ¥, y → + ¥
 * Domain: x Î R +
 * Range: y Î R

Log Graphs with 0 < a < 1


All logarithmic graphs, y = log a x, where 0 < a < 1 have the following in common:
 * Asymptote at x = 0: as x → 0, y → + ¥
 * x-intercept at (1, 0): log a 1 = 0
 * 2nd point at (a, 1): log a a = 1
 * Strictly Decreasing Graph
 * As x → + ¥, y → – ¥
 * Domain: x Î R +.
 * Range: y Î R

Note that: math \log_{0.5}x = -\log_2x math

so using the transformation rules, we can see that: y = log 0.5 x is the reflection of y = log 2 x across the x-axis (in the y-direction)

Transformations
We can use the standard transformations such as dilations, reflections and translations on the y = log a x graph.

Log Graphs on the Classpad


Most calculators have two bases of logarithm built into them.
 * log 10 x (usually represented as **log**)
 * log e x (usually represented as **ln**)

To draw a logarithmic graph with any other base, we would have to use the change of base log law:

** Example 1 **
Sketch on the classpad: y = log 2 x math y = \log_2x=\dfrac{\log_{10}x}{\log_{10}2} math

Note: The Classpad __can__ enter logs with different bases directly {in the virtual keyboard, **2D** tab}

** Example 2 **
Sketch the graph of y = log 3 (2x – 3)

Asymptote occurs at y = log 3 (0) {log 3 (0) is undefined} 2x – 3 = 0 2x = 3 x = 1½ {Equation of Asymptote: x = 1½}
 * Method 1 **

x-intercept occurs at y = log 3 (1) {log 3 (1) = 0} 2x – 3 = 1 2x = 4 x = 2 {x-intercept: (2, 0) }

A 2nd point occurs at y = log 3 (3) {log 3 (3) = 1} 2x – 3 = 3 2x = 6 x = 3 {2nd point: (3, 1) }


 * Method 2 **

y = log 3 (2x – 3)

can be written as: math y = \log_3 \big( 2(x - 1\frac{1}{2} ) \big) math

This is the y = log 3 (x) graph with transformations:
 * dilated by a factor of ½ from the y-axis (in the x-direction) { ½ × x}
 * translated by 1½ units to the right {x + 1½}

In matrix form, this is: math \left[ \begin{matrix} x' \\ y' \\ \end{matrix} \right] = \left[ \begin{matrix} \frac{1}{2} & 0 \\ 0 & 1 \\ \end{matrix} \right] \; \left[ \begin{matrix} x \\ y \\ \end{matrix} \right] + \left[ \begin{matrix} 1\frac{1}{2} \\ 0 \\ \end{matrix} \right] math OR math \\ x'=\frac{1}{2}x+1\frac{1}{2} \\ \\ y' = y math

Asymptote: x = 0 **®** x' = 1½

x-intercept: (x = 1, y = 0) **®** (x' = 2, y' = 0)

2nd point: (x = 3, y = 1) **®** (x' = 3, y' = 1)

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