6.7 Integrals, Exponential Functions, and Logarithms
3 min read•june 24, 2024
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
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General logarithm (base b) to natural logarithm: logb(x)=ln(b)ln(x) (change of base formula)
Natural logarithm to general logarithm: ln(x)=ln(b)⋅logb(x)
General exponential function (base b) to : bx=exln(b)
Natural exponential function to general exponential function: ex=bln(b)x
Example: log2(8)=ln(2)ln(8)≈3 and eln(5)=5
Applications of logarithmic integrals
Exponential growth and decay problems modeled using A(t)=A0ekt
A0: initial amount, k: growth or decay rate, t: time
Total amount over a time interval: ∫t1t2A0ektdt
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: ∫05100e−0.069tdt≈71.65 grams
Behavior analysis through integration
Integral of logarithmic function ∫ln(x)dx=xln(x)−x+C
Analyzes area under the curve or average value over an interval
Integral of exponential function ∫exdx=ex+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: ∫1eln(x)dx≈0.37 and ∫1eexdx≈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 dtdP=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: ∫031000e0.693tdt≈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