6.7 Integrals, Exponential Functions, and Logarithms

Logarithms and exponential functions are essential tools in calculus. They model growth, decay, and other natural phenomena, making them crucial for understanding real-world applications. Their unique properties and relationships allow for powerful mathematical manipulations.

Mastering these functions opens doors to solving complex problems in science, engineering, and finance. From population dynamics to compound interest, logarithms and exponentials provide a framework for analyzing and predicting change over time.

Logarithms and Exponential Functions

Definition of natural logarithm

Top images from around the web for Definition of natural logarithm

• 

  Inverse Functions: Exponential, Logarithmic, and Trigonometric Functions | Boundless Calculus View original

  Is this image relevant?

• 

  Natural logarithm - Wikipedia View original

  Is this image relevant?

• 

  Integrals Involving Exponential and Logarithmic Functions · Calculus View original

  Is this image relevant?

• 

  Inverse Functions: Exponential, Logarithmic, and Trigonometric Functions | Boundless Calculus View original

  Is this image relevant?

• 

  Natural logarithm - Wikipedia View original

  Is this image relevant?

1 of 3

Top images from around the web for Definition of natural logarithm

• 

  Inverse Functions: Exponential, Logarithmic, and Trigonometric Functions | Boundless Calculus View original

  Is this image relevant?

• 

  Natural logarithm - Wikipedia View original

  Is this image relevant?

• 

  Integrals Involving Exponential and Logarithmic Functions · Calculus View original

  Is this image relevant?

• 

  Inverse Functions: Exponential, Logarithmic, and Trigonometric Functions | Boundless Calculus View original

  Is this image relevant?

• 

  Natural logarithm - Wikipedia View original

  Is this image relevant?

1 of 3

• Defined as the integral of $\frac{1}{t}$ from 1 to $x$, expressed as $\ln(x) = \int_1^x \frac{1}{t} dt$ (definite integral)
• Inverse function of the exponential function $e^x$, where $e$ is a mathematical constant approximately equal to 2.71828
• Relationship between $e$ and natural logarithm: $e^{\ln(x)} = x$ and $\ln(e^x) = x$
  • Example: $e^{\ln(5)} = 5$ and $\ln(e^3) = 3$

Differentiation of logarithmic functions

• Derivative of natural logarithm: $\frac{d}{dx} \ln(x) = \frac{1}{x}$
• Derivative of exponential function $e^x$: $\frac{d}{dx} e^x = e^x$
• Integral of $\frac{1}{x}$: $\int \frac{1}{x} dx = \ln|x| + C$, where $C$ is the constant of integration (indefinite integral)
• Integral of $e^x$: $\int e^x dx = e^x + C$, where $C$ is the constant of integration
  • Example: $\frac{d}{dx} \ln(x^2) = \frac{2}{x}$ and $\int e^{3x} dx = \frac{1}{3}e^{3x} + C$

Properties in integral calculations

• Logarithm properties for simplifying expressions:
  1. $\ln(ab) = \ln(a) + \ln(b)$
  2. $\ln(\frac{a}{b}) = \ln(a) - \ln(b)$
  3. $\ln(a^b) = b\ln(a)$
• Exponential function properties for simplifying expressions:
  1. $e^{a+b} = e^a \cdot e^b$
  2. $e^{a-b} = \frac{e^a}{e^b}$
  3. $(e^a)^b = e^{ab}$
• Properties applied before integrating or after differentiating to simplify calculations
  • Example: $\int \ln(x^3) dx = \int (3\ln(x)) dx = 3\int \ln(x) dx = 3(x\ln(x) - x) + C$

Conversion between logarithm types

• General logarithm (base $b$) to natural logarithm: $\log_b(x) = \frac{\ln(x)}{\ln(b)}$ (change of base formula)
• Natural logarithm to general logarithm: $\ln(x) = \ln(b) \cdot \log_b(x)$
• General exponential function (base $b$) to natural exponential function: $b^x = e^{x\ln(b)}$
• Natural exponential function to general exponential function: $e^x = b^{\frac{x}{\ln(b)}}$
  • Example: $\log_2(8) = \frac{\ln(8)}{\ln(2)} \approx 3$ and $e^{\ln(5)} = 5$

Applications of logarithmic integrals

• Exponential growth and decay problems modeled using $A(t) = A_0 e^{kt}$
  • $A_0$: initial amount, $k$: growth or decay rate, $t$: time
  • Total amount over a time interval: $\int_{t_1}^{t_2} A_0 e^{kt} dt$
• Logarithmic functions model situations with diminishing returns or logarithmic scales (Richter scale for earthquakes, pH scale for acidity)
• Integrating logarithmic functions finds area under the curve or average value over an interval
  • Example: Radioactive decay of a substance with half-life of 10 years, initial amount of 100 grams, total amount remaining after 5 years: $\int_0^5 100e^{-0.069t} dt \approx 71.65$ grams

Behavior analysis through integration

• Integral of logarithmic function $\int \ln(x) dx = x\ln(x) - x + C$
  • Analyzes area under the curve or average value over an interval
• Integral of exponential function $\int e^x dx = e^x + C$
  • Analyzes area under the curve or average value over an interval
• Comparing integrals of logarithmic and exponential functions provides insights into relative growth rates and behavior
  • Example: $\int_1^e \ln(x) dx \approx 0.37$ and $\int_1^e e^x dx \approx 6.39$, showing exponential function grows much faster than logarithmic function

Models of exponential growth

• Exponential growth and decay models developed by solving differential equations
  • Population growth modeled by $\frac{dP}{dt} = kP$, where $P$ is population and $k$ is growth rate
• Solution to differential equation found by integrating both sides, resulting in an exponential function
• Integral of exponential function over a specific time interval represents total growth or decay during that period
• Interpreting integral results provides insights into overall behavior of the modeled system
  • Example: Bacterial population starting with 1000 cells, doubling every hour, total population after 3 hours: $\int_0^3 1000e^{0.693t} dt \approx 6646$ cells

Integration techniques for logarithmic and exponential functions

• Antiderivative: A function whose derivative is the given function
• Integration by substitution: Used when integrating composite functions
• Integration by parts: Useful for integrating products of functions, especially with logarithms
• Fundamental Theorem of Calculus: Connects differentiation and integration, crucial for evaluating definite integrals