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Section 5.1 A Calculus Definition for Logarithms

This section presents materials that explain or enable or use the following standards
  • Calculate exponential derivatives
  • Calculate trigonometric derivatives
  • Calculate derivatives with the chain rule
  • Integrate polynomial, trig, and/or exponential functions
We have memorized derivatives for \(ln(x)\) and \(e^x\text{.}\) However no definition or proof was provided. We have also memorized that \(a^x \equiv e^{x \ln a}\) (used in indeterminate form limits). This definition benefits from a context. All of this will be provided in the next sections.

Subsection 5.1.1 Old and New Definitions

Recall that in algebra classes we use \(\log_b(a) = x\) if and only if \(b^x=a\text{.}\)

Activity 5.1.1. Discovering a limitation of a log definition.

The goal of this activity is to recognize why the defintion of logarithms used in algebra courses is insufficient for most uses.

(a)

What is \(\log_2(8)\text{?}\)

(b)

What is \(\log_3(9)\text{?}\)

(c)

What is \(\log_5(5)\text{?}\)

(d)

What is \(\log_5(10)\text{?}\)

(e)

In general what values of logarithms can we evaluate with this definition?

Definition 5.1.2. Natural Logarithm.

\(\ln x = \int_1^x \frac{1}{t} \; dt\)

Subsection 5.1.2 Proving Log Properties

While you are likely already aware of a variety of properties of logarithms, they were demonstrated previously using the algebra definition. Here we show the properties still work using the calculus definition. This provides an opportunity to review multiple concepts from calculus.

Activity 5.1.3. Discovering Increasing/Decreasing.

The goal of this activity is to determine where \(\ln(x)\) is increasing and decreasing.
This activity reviews concepts of integration and increasing functions.

(a)

Recall that \(\ln(x)\) integrates (adds) \(1/t\) for \(t \ge 0\text{.}\) Are the values added positive or negative?

(b)

Note \(\ln(5)=\int_1^5 1/t \; dt\) integrates farther than \(\ln(3)=\int_1^3 1/t \; dt\) because it integrates past 3 to 5. As a result of the previous question’s result is something added or subtracted when integrating farther?

(c)

Which is bigger: \(\ln(2)\) or \(\ln(3)\text{?}\) You can use Figure 5.1.4 to visualize this.

(d)

In general is \(\ln(x)\) or \(\ln(x+a)\) for \(a > 0\) bigger?

(e)

Assuming this works left of zero as well (it does), where is \(\ln(x)\) increasing?
Figure 5.1.4. Natural Log Comparison

Checkpoint 5.1.5. Calculate \(\ln(1)\).

Use the definition of \(\ln x\) to calculate \(\ln(1)\text{.}\) Note how this derives from an integral property.

Activity 5.1.6. Discovering a Product Property.

The goal of this activity is to show a property of logs of products.
This activity uses our knowledge of derivatives of natural log and the chain rule.

(a)

For \(f(x)=\ln(x)\) calculate \(f^\prime(x)\text{.}\)

(b)

For \(g(x)=\ln(ax)\) calculate \(g^\prime(x)\text{.}\)

(d)

Evaluate your conclusion at \(x=1\) to calculate \(C\text{.}\) Plug this into the conclusion.

Activity 5.1.7. Discovering a Power Property.

The goal of this activity is to show a property of logs of powers.

(a)

Use the product property of logs to re-write \(\ln(x^2)\text{.}\)

(b)

Use the product property of logs to re-write \(\ln(x^3)\text{.}\)

(c)

Use the product property of logs to re-write \(\ln(x^5)\text{.}\)

(d)

In general \(\ln(x^n)\) equals what for positive integers \(n\text{?}\)

Subsection 5.1.3 More Trig Anti-Derivatives

Activity 5.1.8. Discovering Four Trig Anti-Derivatives.

The goal of this activity is to discover four trigonometric anti-derivatives.
To be effective you need to calculate these by hand. You will learn nothing by using technology. The first two are practice using the chain rule and the derivative of the natural logarithm. Frequently, these questions are missed in testing: practice to avoid this fate. The next two will be obvious once you have completed the first two. For the last two, make a guess based on the previous (middle) two and check using technology.

(a)

Calculate the derivative of \(f(x)=-\ln(\cos x)\text{.}\)

(b)

Calculate the derivative of \(f(x)=\ln(\sec x + \tan x)\text{.}\)

(c)

What is \(\int \tan x \; dx\text{?}\)

(d)

What is \(\int \sec x \; dx\text{?}\)

(e)

What is \(\int \cot x \; dx\text{?}\)

(f)

What is \(\int \csc x \; dx\text{?}\)
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