Exponential function & Logarithmic function
Exponential function:
- An
exponential function is a mathematical function of the form f(x) =
ab^x, where a and b are constants and b is the base of the exponential
function.
- Exponential functions exhibit exponential growth or decay, depending on whether b is greater than 1 or less than 1.
- The exponential function is used in many areas of mathematics and science, including calculus, finance, physics, and biology.
- The natural exponential function, denoted by e^x, is the exponential function with the base 'e'.
- The
exponential function is the inverse of the logarithmic function, and
they can be used to solve equations involving logarithmic and
exponential functions.
Logarithmic function:
- A logarithmic
function is a mathematical function of the form f(x) = logb(x), where b
is the base of the logarithmic function and x is the argument.
- Logarithmic functions are used to express the relationship between two quantities that are being multiplied or divided.
- The natural logarithmic function, denoted by ln(x), is the logarithmic function with the base 'e'.
- Logarithmic functions have several properties, including the product rule, quotient rule, power rule, and change of base rule.
- Logarithmic
functions are the inverse of exponential functions, and they can be
used to solve equations involving exponential and logarithmic functions.
Important Formulas to Remember in these Exponential function & Logarithmic function
Exponential Function:
- Definition: $$e^x = \sum_{n=0}^{\infty} \frac{x^n}{n!}$$
- Properties:
- $$e^{a+b} = e^ae^b$$
- $$\frac{d}{dx}e^x = e^x$$
- $$\int e^x\,dx = e^x + C$$
Logarithmic Function:
- Definition: $$\ln x = \log_e x$$
- Properties:
- $$\ln 1 = 0$$
- $$\ln e = 1$$
- $$\ln(xy) = \ln x + \ln y$$
- $$\ln\left(\frac{x}{y}\right) = \ln x - \ln y$$
- $$\ln(x^c) = c\ln x$$
- $$\ln e^x = x$$
- $$\frac{d}{dx}\ln x = \frac{1}{x}$$
- $$\int \frac{1}{x}\,dx = \ln |x| + C$$