We can use the inverse relationship between logarithmic and exponential functions to explore the graphs and characteristics of the parent logarithmic functions, including natural logarithmic functions.

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The graph of f\left(x\right)=b^x has a point at \left(0,1\right), a point at \left(1,b\right), and an asymptote at y=0.

Since f\left(x\right)=\log_b\left(x\right) is the inverse of f\left(x\right)=b^x, and an inverse function is a reflection across the line y = x which maps each point \left(x, y\right) to \left(y, x\right), its graph has a point at \left(1,0\right) and \left(b,1\right) and an asymptote of x=0.

The points that were approaching the y-axis on the parent exponential function are now approaching the x-axis in the logarithmic function. This means the parent logarithmic function will have a vertical asymptote, in contrast to the parent exponential function which has a horizontal asymptote.

Examples

Example 1

Consider the function f\left(x\right)=\log_{\frac{1}{4}}x.

State the domain and range.

Worked Solution

Create a strategy

The function has not been translated, so the domain and range will be the same as the parent logarithmic function.

Apply the idea

Domain: \left(0,\infty\right)

Range: \left(-\infty,\infty\right)

Sketch a graph of the function.

Worked Solution

Create a strategy

Since 0<b<1, we know that this is a decreasing logarithmic function. The inverse of this function is y=\left(\frac{1}{4}\right)^x. We can use the inverse to find key points, then graph the given function by switching the x- and y-values of the key points.

Apply the idea

We will begin by making a table of values for y=\left(\frac{1}{4}\right)^x.

x	-2	-1	0	1	2
y	16	4	1	\frac{1}{4}	\frac{1}{16}

Because f\left(x\right)=\log_{\frac{1}{4}}x is the inverse, switching the rows of the above table creates a table of values for f\left(x\right).

x	16	4	1	\frac{1}{4}	\frac{1}{16}
f\left(x\right)	-2	-1	0	1	2

Now, we can use the table to graph the function.

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Reflect and check

Although we could have used a calculator to build the table of values, using the inverse relationship allowed us to find nice points instead of estimating decimals on the graph.

Example 2

Consider the graph of the logarithmic function f\left(x\right).

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Determine the equation of the asymptote and two points on the curve.

Worked Solution

Create a strategy

The asymptote is displayed visually as a dashed line that the function approaches, but does not cross. On this graph, it is a vertical line which means the equation will be in the form x=c.

Apply the idea

The equation of the asymptote is x=-4 and the graph has points at \left(-3,0\right) and \left(-2,1\right).

Reflect and check

There are also points located at \left(0,2\right) and \left(4,3\right).

Sketch the inverse function on the same coordinate plane.

Worked Solution

Create a strategy

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We can graph the inverse relation by reflecting f\left(x\right) across the line y=x.

We can use the key points, that have integer values, at \left(-3, 0\right), \left(-2, 1\right), \left(0, 2\right), and \left(4, 3\right) and swap the x- and y-values.

\left(-3, 0\right) \to \left(0, -3\right)
\left(-2, 1\right) \to \left(1, -2\right)
\left(0, 2\right) \to \left(2, 0\right)
\left(4, 3\right) \to \left(3, 4\right)

This allows us to approximate a few more points on the inverse function, then sketch the curve through these inverted points.

The asymptote must also be reflected across the line y=x. This takes x=-4 to y=-4.

Apply the idea

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Write the equation of the inverse function.

Worked Solution

Create a strategy

As f\left(x\right) is a logarithmic function, the inverse will be an exponential function in the form y=ab^{\left(x-h\right)}+k

We will use the graph from part (b) to identify any transformations that were applied to the parent exponential function, y=b^x.

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We can use the key features of an exponential function to determine the transformations and values of a, h, and k, then analyze the change in the y-values to find the value of b.

Apply the idea

We can see the asymptote is at y=-4, which indicates a vertical translation of 4 units down has occurred. This tells us k=-4. Since the parent function would have had a y-intercept at y=1, translating it down 4 units would make the new y-intercept at y=-3.

The y-intercept in our exponential graph is at y=-3 which indicates there is no stretch factor, so a=1. It also means that there has not been a horizontal shift, indicating h=0.

Since this function has been transformed, we cannot divide the outputs to determine the constant factor, b. By analyzing the change in y-values we can determine the value of the base, b.

x	y	\text{Change in }y
0	-3
1	-2	\text{Increases by }1
2	0	\text{Increases by }2
3	4	\text{Increases by }4

Since the change in y is doubling each time, the base of the exponential function is 2.

Therefore, the inverse function is f^{-1}\left(x\right)=2^{x}-4.

Reflect and check

We can confirm this is the correct function by testing some of the other points:

x	2^{x}-4	y
0	2^0-3	-3
1	2^1-4	-2
2	2^2-4	0
3	2^3-4	4

Based on the graph, these are the expected values.

Idea summary

We can use the inverse relationship between logarithmic and exponential functions to find key points with which to sketch the graph of a logarithmic function.

Outcomes

2.10.A

Construct representations of the inverse of an exponential function with an initial value of 1.

2.10 Inverses of exponential functions

Introduction

Ideas

Logarithmic functions

Examples

Example 1

Example 2

Outcomes

2.10.A

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