Mathematical Analysis Notes - Exponential and Logarithmic Functions

MODULE III: EXPONENTIAL AND LOGARITHMIC FUNCTIONS

1. EXPONENTIAL FUNCTIONS

  • Defined as f(x)=Abxf(x) = Ab^x (where A and b are constants, b > 0).

  • The base ee is an important constant defined by:
    e=extlimnoext∞(1+rac1n)ne = ext{lim}_{n o ext{∞}} \big(1 + rac{1}{n}\big)^n
    (Approximation of e values as n increases).

  • Common exponential function: f(x)=exf(x) = e^x; graph lies between 2x2^x and 3x3^x.

2. DOMAIN, RANGE, AND ASYMPTOTES

  • For f(x)=exf(x) = e^x:

    • Domain: (−∞,∞)(-∞, ∞)

    • Range: (0,∞)(0, ∞)

    • Horizontal asymptote: y=0y=0

  • Transformed functions shift graph and change asymptotes accordingly.

3. LAWS OF EXPONENTS

  • Product Rule: amimesan=am+na^m imes a^n = a^{m+n}

  • Quotient Rule: racaman=am−nrac{a^m}{a^n} = a^{m-n}

  • Power Rule: (am)n=amn(a^m)^n = a^{mn}

4. EVALUATING EXPONENTIAL EXPRESSIONS

  • Simplification procedures involve applying the laws of exponents.

  • Example techniques for evaluating include direct calculation and the use of a calculator.

5. SOLVING EXPONENTIAL EQUATIONS

  • To solve equations of the form ax=ba^x = b, apply logarithms to both sides:
    x=racextloga(b)1x = rac{ ext{log}_a(b)}{1}

  • Example: Solve 2x=82^x = 8 -> x=3x = 3.

6. LOGARITHMIC FUNCTIONS

  • Defined as the inverse of exponential functions: if y=axy = a^x then x=extloga(y)x = ext{log}_a(y).

  • Characteristics: Domain is x > 0, range is all real numbers, with a vertical asymptote at x=0x=0.

7. LOGARITHMIC PROPERTIES

  • Logarithm of a product: extlog<em>a(MN)=extlog</em>a(M)+extloga(N)ext{log}<em>a(MN) = ext{log}</em>a(M) + ext{log}_a(N)

  • Logarithm of a quotient: extlog<em>a(racMN)=extlog</em>a(M)−extloga(N)ext{log}<em>a( rac{M}{N}) = ext{log}</em>a(M) - ext{log}_a(N)

  • Logarithm of a power: extlog<em>a(Mr)=rextlog</em>a(M)ext{log}<em>a(M^r) = r ext{log}</em>a(M)

8. SOLVING LOGARITHMIC EQUATIONS

  • Convert logarithmic equations to exponential form for solving.
    Example: extlogb(x)=yext{log}_b(x) = y implies x=byx = b^y.

  • Solve for variable xx by isolating it.

9. TRANSFORMING LOGARITHMIC EXPRESSIONS

  • Logarithmic expressions can be consolidated or expanded using properties.

  • Change of base formula for logarithmic evaluations not in standard bases.

10. FINAL NOTES

  • The relationship between exponential and logarithmic functions is foundational in mathematical analysis.

  • Understanding the transformations and properties enhances graphing and solving capabilities.