6.3 Logarithmic Functions

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6.3 Logarithmic Functions

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Logarithmic Functions

If b > 0 and b ≠ 1, the exponential function f(x) = b^x is either increasing or decreasing and so it is one-to-one by the

Horizontal Line Test.

It therefore has an inverse function f^–1, which is called the logarithmic function with base b and is denoted by log_b.

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Logarithmic Functions

If we use the formulation of an inverse function, f^–1(x) = y f(y) = x

then we have

Thus, if x > 0, then log_bx is the exponent to which the base b must be raised to give x.

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Evaluate (a) log₃81, (b) log₂₅5, and (c) log₁₀0.001.

Solution:

(a) log₃81 = 4 because 3⁴ = 81 (b) log₂₅5 = because 25^1/2= 5

(c) log₁₀0.001 = –3 because 10^–3 = 0.001

Example 1

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The cancellation equations, when applied to the functions f(x) = b^x and f^–1(x) = log_bx, become

The logarithmic function log_b has domain (0, ) and range and is continuous since it is the inverse of a continuous

function, namely, the exponential function.

Its graph is the reflection of the graph of y = b^x about the line y = x.

Logarithmic Functions

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Figure 1 shows the case where b > 1. (The most important logarithmic functions have base b > 1.)

The fact that y = b^x is a very rapidly increasing function for x > 0 is reflected in the fact that y = log_bx is a very slowly increasing function for x > 1.

Figure 1

Logarithmic Functions

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Figure 2 shows the graphs of y = log_bx with various values of the base b > 1. Since log_b1 = 0, the graphs of all

logarithmic functions pass through the point (1, 0).

Figure 2

Logarithmic Functions

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The following theorem summarizes the properties of logarithmic functions.

Logarithmic Functions

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In particular, the y-axis is a vertical asymptote of the curve y = log_bx.

Logarithmic Functions

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Find log₁₀(tan²x).

Solution:

As x → 0, we know that t = tan²x → tan²0 = 0 and the

values of t are positive. So by (4) with b = 10 > 1, we have

Example 3

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Natural Logarithms

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Natural Logarithms

Of all possible bases b for logarithms, we will see that the most convenient choice of a base is the number e.

The logarithm with base e is called the natural logarithm and has a special notation:

If we put b = e and replace log_e with “ln” in (1) and (2), then the defining properties of the natural logarithm function

become

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Natural Logarithms

In particular, if we set x = 1, we get

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Example 4

Find x if ln x = 5.

Solution 1:

From (5) we see that

ln x = 5 means e⁵ = x Therefore x = e⁵.

(If you have trouble working with the “ln” notation, just

replace it by log_e. Then the equation becomes log_ex = 5; so, by the definition of logarithm, e⁵ = x.)

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Example 4 – Solution

Solution 2:

Start with the equation

ln x = 5

and apply the exponential function to both sides of the equation:

e^{ln x} = e⁵

But the second cancellation equation in (6) says that e^{ln x} = x.

Therefore x = e⁵.

cont’d

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Natural Logarithms

The following formula shows that logarithms with any base can be expressed in terms of the natural logarithm.

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Example 7

Evaluate log₈ 5 correct to six decimal places.

Solution:

Formula 7 gives

log₈ 5 =

≈ 0.773976

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Graph and Growth of the Natural

Logarithm

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Graph and Growth of the Natural Logarithm

The graphs of the exponential function y = e^xand its

inverse function, the natural logarithm function, are shown in Figure 3.

Because the curve y = e^x

crosses the y-axis with a slope of 1, it follows that the reflected curve y = ln x crosses the x-axis with a slope of 1.

Figure 3

The graph of y = ln x is the reflection of the graph of y = e^x about the line y = x.

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Graph and Growth of the Natural Logarithm

In common with all other logarithmic functions with base greater than 1, the natural logarithm is a continuous,

increasing function defined on (0, ) and the y-axis is a vertical asymptote.

If we put b = e in (4), then we have the following limits:

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Example 8

Sketch the graph of the function y = ln(x – 2) – 1.

Solution:

We start with the graph of y = ln x as given in Figure 3.

Figure 3

The graph of y = ln x is the reflection of the graph of y = e^x about the line y = x.

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Example 8

Using the transformations, we shift it 2 units to the right to get the graph of y = ln(x – 2) and then we shift it 1 unit

downward to get the graph of y = ln(x – 2) – 1.

(See Figure 4.)

Notice that the line x = 2 is a vertical asymptote since

Figure 4

cont’d

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Graph and Growth of the Natural Logarithm

We have seen that ln x → as x → . But this happens very slowly. In fact, ln x grows more slowly than any

positive power of x.

To illustrate this fact, we compare approximate values of

the functions y = ln x and y = x^1/2 = in the following table.

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Graph and Growth of the Natural Logarithm

We graph them in Figures 5 and 6.

You can see that initially the graphs of y = and y = ln x grow at comparable rates, but eventually the root function far surpasses the logarithm.

Figure 5 Figure 6

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Graph and Growth of the Natural Logarithm

In fact, we will be able to show

for any positive power p.

So for large x, the values of ln x are very small compared with x^p.