2.7 Integrals, Exponential Functions, and Logarithms
4 min read•june 24, 2024
Natural logarithms and exponential functions are key players in calculus. They're like two sides of the same coin, with the natural log being the integral of 1/x and e^x being its own derivative. These functions pop up everywhere in math and science.
Understanding how to work with these functions is crucial. You'll use them to solve complex integrals, model real-world phenomena, and tackle limits. Their unique properties make them indispensable tools for tackling a wide of calculus problems.
Natural Logarithms and Exponential Functions
Definition of natural logarithm
Top images from around the web for Definition of natural logarithm
The ln(x) represents the integral of the reciprocal function t1 from 1 to x, expressing the area under the curve between these limits
The number e (), approximately 2.71828, serves as the base of the natural logarithm and arises from the limit of (1+n1)n as n approaches infinity, representing continuous
The natural ex and natural logarithm ln(x) are , meaning they "undo" each other (eln(x)=x and ln(ex)=x)
Differentiation of logarithmic functions
The derivative of the natural logarithm ln(x) is the reciprocal function x1, indicating that the rate of change of ln(x) is inversely proportional to x
The derivative of the natural exponential function ex is itself ex, demonstrating that ex grows at a rate proportional to its current value
The integral of the reciprocal function x1 is the natural logarithm ln∣x∣+C, where C is the constant of integration ()
The integral of the natural exponential function ex is itself ex+C, where C is the constant of integration
Properties for integral solutions
allow for simplifying expressions: ln(xy)=ln(x)+ln(y) (product rule), ln(yx)=ln(x)−ln(y) (quotient rule), and ln(xn)=nln(x) (power rule)
enable manipulation of terms: ex+y=ex⋅ey (product of exponentials), ex−y=eyex (quotient of exponentials), and (ex)n=enx (power of exponential)
Conversion between logarithm types
To convert a general logarithm with base b to a natural logarithm, divide the natural logarithm of x by the natural logarithm of b: logb(x)=ln(b)ln(x) ()
To convert a natural logarithm to a general logarithm with base b, multiply the natural logarithm of x by the natural logarithm of b: ln(x)=ln(b)⋅logb(x)
To convert a general exponential function with base b to a natural exponential function, raise e to the power of x multiplied by the natural logarithm of b: bx=exln(b)
Integration Techniques and Applications
Integration of logarithmic expressions
Integration by substitution is useful for integrals involving exponential functions (substitute u=ex) or logarithmic functions (substitute u=ln(x))
is suitable for integrals of the form ∫x⋅exdx or ∫ln(x)dx
For ∫x⋅exdx, let u=x and dv=exdx, resulting in x⋅ex−∫exdx=x⋅ex−ex+C
For ∫ln(x)dx, let u=ln(x) and dv=dx, resulting in xln(x)−∫xxdx=xln(x)−x+C
Behavior of logarithmic functions
The natural logarithm ln(x) has a of (0,∞) and a range of (−∞,∞), increasing for all x>0 with a at x=0
The natural exponential function ex has a domain of (−∞,∞) and a range of (0,∞), always increasing with a at y=0 as x approaches −∞
can be applied to evaluate limits involving indeterminate forms of logarithmic and exponential functions
Applications of logarithmic integrals
Exponential growth and decay phenomena (population growth, radioactive decay, compound interest) can be modeled using exponential functions, with the integral representing the total growth or decay over a given time period
, such as the Richter scale (earthquake magnitudes) and decibel scale (sound intensity), utilize logarithmic functions, and integrating these functions can help calculate quantities related to these scales
Fundamental Theorem of Calculus and Types of Integrals
Fundamental Theorem of Calculus
The First establishes the relationship between differentiation and integration, stating that the derivative of a with respect to its upper limit is the integrand evaluated at that limit
The Second Fundamental Theorem of Calculus provides a method for evaluating definite integrals using antiderivatives
Types of Integrals
Indefinite integrals represent the general antiderivative of a function and include a constant of integration
Definite integrals calculate the signed area between a function and the x-axis over a specified interval, using the fundamental theorem of calculus to evaluate