Logarithmic Function Graph

Understanding Logarithmic Function Graphs

Have you ever noticed how sound decreases in intensity as we move away from its source? Or how the pH of a solution changes? These phenomena can be modeled with logarithmic function graphs. Let's learn the characteristics and how to draw logarithmic function graphs.

Characteristics of Logarithmic Graphs

Logarithmic function graphs have a distinctive shape different from other functions. Let's look at the basic graph $y = \log_b x$ for various base values.

Important Properties of Logarithmic Graphs

For function $f(x) = \log_b x$ with $b > 1$:

1. Domain: $x > 0$ (positive numbers only)
2. Range: All real numbers ($-\infty < y < \infty$)
3. x-intercept: $(1, 0)$ because $\log_b 1 = 0$
4. Vertical asymptote: y-axis ($x = 0$)
5. Function behavior:
   • Increasing for $b > 1$
   • Decreasing for $0 < b < 1$

Drawing Logarithmic Function Graphs

Let's learn the steps to draw logarithmic function graphs with concrete examples.

1. Drawing $y = \log_2 x$

   To draw this graph, we create a table of values by choosing $x$ values that are powers of 2:

   $x$	$\frac{1}{8}$	$\frac{1}{4}$	$\frac{1}{2}$	1	2	4	8
   $y = \log_2 x$	-3	-2	-1	0	1	2	3

   Graph of $y = \log_2 x$

   Notice the important points and the curve shape.

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2. Drawing $y = \log_{\frac{1}{3}} x$

   For base $0 < b < 1$, the graph will be decreasing:

   $x$	$\frac{1}{27}$	$\frac{1}{9}$	$\frac{1}{3}$	1	3	9	27
   $y = \log_{\frac{1}{3}} x$	3	2	1	0	-1	-2	-3

   Graph of $y = \log_{\frac{1}{3}} x$

   Graph decreases because the base is less than 1.

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Comparing Logarithmic Graphs

Let's compare logarithmic graphs with different bases on one coordinate system:

Transformations of Logarithmic Graphs

Logarithmic graphs can be transformed in various ways:

Vertical Translation

We can shift the logarithmic function graph by adding or subtracting a constant $k$ to the function.

Horizontal Translation

We can shift the logarithmic function graph by adding or subtracting a constant $h$ to the function.

Exercises

1. Create a value table and draw the graphs of:

   • $y = \log_3 x$
   • $y = \log_{\frac{1}{2}} x$

2. Determine the domain, range, and asymptote of function $f(x) = \log_5 (x + 3)$.

3. If $f(x) = \log_2 x$ and $g(x) = \log_2 (x - 4)$, determine:

   • The shift of graph $g(x)$ relative to $f(x)$
   • The domain of $g(x)$

4. Sketch the graph of $y = \log_3 x + 2$ and determine the y-intercept.

Answer Key

1. Value tables:

   For $y = \log_3 x$:

   $x$	$\frac{1}{9}$	$\frac{1}{3}$	1	3	9
   $y$	-2	-1	0	1	2

   For $y = \log_{\frac{1}{2}} x$:

   $x$	$\frac{1}{4}$	$\frac{1}{2}$	1	2	4
   $y$	2	1	0	-1	-2

2. For $f(x) = \log_5 (x + 3)$:

   • Domain: $x + 3 > 0 \Rightarrow x > -3$
   • Range: All real numbers
   • Vertical asymptote: $x = -3$

3. For $g(x) = \log_2 (x - 4)$:

   • Shift: 4 units to the right
   • Domain: $x - 4 > 0 \Rightarrow x > 4$

4. For $y = \log_3 x + 2$:

   • Graph $y = \log_3 x$ shifted 2 units up
   • There is no y-intercept because the domain is $x > 0$
   Sketch of Graph $y = \log_3 x + 2$

   Logarithmic graph base 3 shifted 2 units up.

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