**The Logarithm Function and the Derivative**

Now you will see why our new definition of logarithm is so convenient. If we want to differentiate the logarithm function we can apply the Fundamental Theorem of Calculus:

More generally,

**Example 1**

Calculate

**Solution 1**

For the first problem, we let *u* = 4 + *x* and *du/dx* = 1. Therefore we have

Similarly,

**You Try It** : What is the derivative of the function ln( *x*^{3} + *x*^{2} )?

Now we examine the graph of *y* = ln *x*. Since

we know that ln *x* is an increasing, concave down function whose graph passes through (1,0). There are no relative maxima or minima (since the derivative is never 0). Certainly ln .9 < 0; the formula ln(.9 ^{n} ) = *n* ln .9 therefore tells us that ln *x* is negative without bound as *x* → 0 ^{+} . Since ln *x* = − ln(1/ *x* ), we may also conclude that ln *x* is positive without bound as *x* → +∞. A sketch of the graph of *y* = ln *x* appears in Fig. 6.2.

We learned in the last paragraph that the function ln *x* takes negative values which are arbitrarily large in absolute value when *x* is small and positive. In particular, the negative *y* axis is a vertical asymptote. Since ln(1/ *x* ) = − ln *x* , we then find that ln *x* takes arbitrarily large positive values when *x* is large and positive. The graph exhibits these features.

Since we have only defined the function ln *x* when *x* > 0, the graph is only sketched in Fig. 6.2 to the right of the *y* -axis. However it certainly makes sense to discuss the function ln | *x* | when *x* ≠ 0 (Fig. 6.3):

If *x* ≠ 0 then

In other words,

More generally, we have

and

**Example 2**

Calculate

**Solution 2**

**You Try It** : Calculate the integral

**You Try It** : Calculate the integral

**Example 3**

Evaluate the integral

**Solution 3 **

For clarity we set φ( *x* ) = 3 sin *x* −4, φ′( *x* ) = 3(cos *x* ). The integral then has the form

Resubstituting the expression for φ( *x* ) yields that

**You Try It** : Evaluate .

**Example 4**

Calculate

∫ cot *x dx*.

**Solution 4**

We rewrite the integral as

For clarity we take φ(*x*) = sin *x* , φ′( *x* ) = cos *x*. Then the integral becomes

Resubstituting the expression for φ yields the solution:

∫ cot *x dx* = ln |sin *x*| + *C*.

Find practice problems and solutions for these concepts at: Transcendental Functions Practice Test.

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