6.7 Integrals, Exponential Functions, and Logarithms

Logarithms and Exponential Functions

The natural logarithm and the exponential function show up constantly in calculus because they have uniquely clean derivatives and integrals. This section covers how to differentiate, integrate, and apply these functions, along with the properties that make them so useful for modeling growth and decay.

Definition of the Natural Logarithm

The natural logarithm is defined as an integral:

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

This means $\ln(x)$ equals the area under the curve $\frac{1}{t}$ from $t = 1$ to $t = x$. When $x > 1$, the area is positive. When $0 < x < 1$, the integral goes "backwards," so $\ln(x)$ is negative.

The natural logarithm is the inverse of the exponential function $e^x$, where $e \approx 2.71828$. That inverse relationship gives you two identities worth memorizing:

• $e^{\ln(x)} = x$ for $x > 0$
• $\ln(e^x) = x$ for all $x$

For example, $e^{\ln(5)} = 5$ and $\ln(e^3) = 3$. These come up constantly when you need to "undo" a logarithm or an exponential.

Differentiation and Integration Rules

These four results are the core formulas for this section:

• $\frac{d}{dx} \ln(x) = \frac{1}{x}$
• $\frac{d}{dx} e^x = e^x$
• $\int \frac{1}{x}\, dx = \ln|x| + C$
• $\int e^x\, dx = e^x + C$

Notice the absolute value in $\ln|x| + C$. That's there because $\frac{1}{x}$ is defined for negative $x$ too, but $\ln(x)$ only takes positive inputs. The absolute value extends the antiderivative to negative values of $x$.

With the chain rule, these extend naturally. For example:

• $\frac{d}{dx} \ln(x^2) = \frac{2x}{x^2} = \frac{2}{x}$
• $\int e^{3x}\, dx = \frac{1}{3}e^{3x} + C$ (using substitution with $u = 3x$)

Properties for Simplifying Calculations

Before integrating or differentiating, you can often simplify using logarithm and exponential properties. This can save you from harder techniques like integration by parts.

Logarithm properties:

1. $\ln(ab) = \ln(a) + \ln(b)$

2. $\ln\!\left(\frac{a}{b}\right) = \ln(a) - \ln(b)$

3. $\ln(a^b) = b\ln(a)$

Exponential properties:

1. $e^{a+b} = e^a \cdot e^b$
2. $e^{a-b} = \frac{e^a}{e^b}$
3. $(e^a)^b = e^{ab}$

Here's a good example of simplifying before integrating. Suppose you need $\int \ln(x^3)\, dx$. Use property 3 to pull the exponent out first:

$\int \ln(x^3)\, dx = \int 3\ln(x)\, dx = 3\int \ln(x)\, dx = 3(x\ln(x) - x) + C$

That last step uses the standard result $\int \ln(x)\, dx = x\ln(x) - x + C$, which comes from integration by parts.

Conversion Between Logarithm and Exponential Types

In calculus, you almost always want to work with $\ln$ and $e^x$ rather than other bases. The change-of-base formula lets you convert:

• General log to natural log: $\log_b(x) = \frac{\ln(x)}{\ln(b)}$
• General exponential to natural exponential: $b^x = e^{x\ln(b)}$

The second formula is especially useful. For instance, to differentiate or integrate $2^x$, rewrite it as $e^{x\ln(2)}$, and now you can use the standard $e^x$ rules.

Example: $\log_2(8) = \frac{\ln(8)}{\ln(2)} = \frac{\ln(2^3)}{\ln(2)} = \frac{3\ln(2)}{\ln(2)} = 3$

Applications of Logarithmic and Exponential Integrals

Exponential growth and decay are modeled by:

$A(t) = A_0 e^{kt}$

where $A_0$ is the initial amount, $k$ is the growth/decay rate, and $t$ is time. When $k > 0$, you have growth; when $k < 0$, decay.

To find the total accumulated quantity over a time interval, integrate:

$\int_{t_1}^{t_2} A_0 e^{kt}\, dt = \frac{A_0}{k}\left(e^{kt_2} - e^{kt_1}\right)$

Example: A radioactive substance has a half-life of 10 years and an initial amount of 100 grams. The decay rate is $k = \frac{\ln(1/2)}{10} \approx -0.0693$. To find the total gram-years (accumulated amount) over the first 5 years:

$\int_0^5 100e^{-0.0693t}\, dt = \frac{100}{-0.0693}\left(e^{-0.0693(5)} - e^0\right) \approx \frac{100}{-0.0693}(0.707 - 1) \approx 422.8 \text{ gram-years}$

Behavior Analysis Through Integration

Comparing definite integrals of logarithmic and exponential functions reveals how differently they grow.

• $\int \ln(x)\, dx = x\ln(x) - x + C$
• $\int e^x\, dx = e^x + C$

Over the interval $[1, e]$:

• $\int_1^e \ln(x)\, dx = [x\ln(x) - x]_1^e = (e \cdot 1 - e) - (1 \cdot 0 - 1) = 1$
• $\int_1^e e^x\, dx = e^e - e^1 \approx 15.09 - 2.72 \approx 12.37$

The exponential integral is much larger, which reflects how rapidly $e^x$ grows compared to the slow increase of $\ln(x)$.

Models of Exponential Growth

Many growth and decay models start from the differential equation:

$\frac{dP}{dt} = kP$

Solving this by separating variables and integrating both sides:

1. Separate: $\frac{dP}{P} = k\, dt$
2. Integrate: $\int \frac{dP}{P} = \int k\, dt$
3. Result: $\ln|P| = kt + C_1$
4. Exponentiate: $P = e^{kt + C_1} = e^{C_1} \cdot e^{kt} = P_0 e^{kt}$

where $P_0$ is the initial population at $t = 0$.

Example: A bacterial population starts at 1000 cells and doubles every hour, so $k = \ln(2) \approx 0.693$. The total accumulated cell-hours over 3 hours:

$\int_0^3 1000e^{0.693t}\, dt = \frac{1000}{0.693}\left(e^{0.693(3)} - 1\right) = \frac{1000}{0.693}(8 - 1) \approx 10,101 \text{ cell-hours}$

Note that $e^{0.693 \times 3} = e^{3\ln 2} = 2^3 = 8$, which confirms the population doubles each hour (reaching 8000 at $t = 3$).

Integration Techniques for Logarithmic and Exponential Functions

Several techniques come together in this section. Here's when to use each:

• Direct antiderivatives: Use $\int e^x\, dx = e^x + C$ and $\int \frac{1}{x}\, dx = \ln|x| + C$ whenever the integrand matches these forms directly.
• Substitution ($u$-sub): Use when you have a composite function. For example, $\int \frac{2x}{x^2 + 1}\, dx$ works with $u = x^2 + 1$, giving $\ln|x^2 + 1| + C$.
• Integration by parts: Use for products involving $\ln(x)$. The standard choice is to let $u = \ln(x)$ and $dv$ be everything else. This is how you derive $\int \ln(x)\, dx = x\ln(x) - x + C$.
• Fundamental Theorem of Calculus: Connects everything. To evaluate $\int_a^b f(x)\, dx$, find an antiderivative $F(x)$ and compute $F(b) - F(a)$.