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# Simple Exponent and Logarithm Equations Help (page 2)

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By McGraw-Hill Professional
Updated on Oct 4, 2011

#### Examples

Sketch the graph of the logarithmic functions.

• y = log 2 x
• Rewrite the equation in exponential form, x = 2 y , and let the exponent, y , be the numbers −3, −2, −1, 0, 1, 2, and 3.

Table 9.6

 x y −3 −2 −1 1 0 2 1 4 2 8 3

Fig. 9.9

• y = ln x
• Rewritten as an exponent equation, this is x = e y. Let y = −3, −2, −1, 0, 1, 2, and 3.

Table 9.7

 x y 0.05 −3 0.14 −2 0.37 −1 1 0 2.72 1 7.39 2 20.09 3

Fig. 9.10

As you can see by these graphs, the domain of the function f ( x ) = log a x is all positive real numbers, (0, ∞).

### Graphing Simple Exponent and Logarithm Equations Practice Problems

#### Practice

Sketch the graph of the logarithmic function.

1. y = log 1.5 x
2. y = log 3 x

1.

Fig. 9.11

2.

Fig. 9.12

## Finding the Domain of a Function

As long as a is larger than 1, all graphs for f ( x ) = log a x look pretty much the same. The larger a is, the flatter the graph is to the right of x = 1. Knowing this and knowing how to graph transformations, we have a good idea of the graphs of many logarithmic functions.

• The graph of f ( x ) = log 2 ( x − 2) is the graph of y = log 2 x shifted to the right 2 units.
• The graph of f ( x ) = − 5 + log 3 x is the graph of y = log 3 x shifted down 5 units.
• log x is the graph of y = log x flattened vertically by a factor of one-third.

The domain of f ( x ) = log a x is all positive numbers. This means that we cannot take the log of 0 or the log of a negative number. The reason is that a is a positive number. Raising a positive number to any power is always another positive number.

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