Graphing Exponential Functions and Properties

Graphing Exponential Functions

• Basic Form: ( f(x) = a \cdot b^x )
  • ( a ): vertical stretch/compression.
  • ( b ): base of the exponential (if ( b > 1 ), growth; if ( 0 < b < 1 ), decay).
• Key Features:
  • Y-Intercept: ( (0, a) ).
  • Horizontal Asymptote: ( y = 0 ) (as ( x \to -\infty )).
  • Increasing/Decreasing:
    • Increasing if ( b > 1 ).
    • Decreasing if ( 0 < b < 1 ).
• Domain & Range:
  • Domain: ( (-\infty, \infty) ).
  • Range: ( (0, \infty) ) for ( a > 0 ) or ( (-\infty, 0) ) for ( a < 0 ).

Properties of Exponents

1. Product of Powers: ( a^m \cdot a^n = a^{m+n} )
2. Quotient of Powers: ( a^m / a^n = a^{m-n} )
3. Power of a Power: ( (a^m)^n = a^{m \cdot n} )
4. Power of a Product: ( (ab)^n = a^n \cdot b^n )
5. Power of a Quotient: ( (a/b)^n = a^n / b^n )
6. Zero Exponent: ( a^0 = 1 ) (where ( a \neq 0 ))
7. Negative Exponent: ( a^{-n} = 1/a^n )

Logarithmic Identities

1. Definition: If ( b^y = x ), then ( \log_b(x) = y ).
2. Logarithm of a Product: ( \log_b(xy) = \log_b(x) + \log_b(y) )
3. Logarithm of a Quotient: ( \log_b(x/y) = \log_b(x) - \log_b(y) )
4. Logarithm of a Power: ( \log_b(x^n) = n \cdot \log_b(x) )
5. Change of Base Formula: ( \log_b(a) = \frac{\log_k(a)}{\log_k(b)} )
6. Logarithm of 1: ( \log_b(1) = 0 )
7. Logarithm of the Base: ( \log_b(b) = 1 )

Exponential Growth and Decay

• Exponential Growth:
  • Model: ( P(t) = P_0 \cdot e^{rt} )
  • ( P_0 ): initial amount, ( r ): growth rate, ( t ): time.
• Exponential Decay:
  • Model: ( P(t) = P_0 \cdot e^{-rt} )
  • Characterized by a constant percentage decrease.
• Half-Life:
  • Time required for a quantity to reduce to half its initial value, commonly used in radioactive decay.

Applications of Logarithms

• Solving Exponential Equations: Use logarithmic identities to isolate the variable.
• pH Scale: ( \text{pH} = -\log_{10}[\text{H}^+] ) measures acidity.
• Richter Scale: Measures earthquake intensity logarithmically.
• Sound Intensity: Decibels also use logarithmic scales: ( dB = 10 \log_{10}(I/I_0) ).
• Finance: Compound interest calculations can involve logarithmic functions for time in growth models.

Graphing Exponential Functions

• Basic form of an exponential function: ( f(x) = a \cdot b^x ).
• Coefficient ( a ) indicates the vertical stretch or compression of the graph.
• Base ( b ) determines the nature of the function:
  • If ( b > 1 ), the function exhibits exponential growth.
  • If ( 0 < b < 1 ), it shows exponential decay.
• Key Features:
  • Y-intercept is located at ( (0, a) ).
  • The horizontal asymptote is established at ( y = 0 ) as ( x \to -\infty ).
  • The function is increasing for ( b > 1 ) and decreasing for ( 0 < b < 1 ).
• Domain & Range:
  • Domain covers all real numbers: ( (-\infty, \infty) ).
  • Range is ( (0, \infty) ) if ( a > 0 ) and ( (-\infty, 0) ) if ( a < 0 ).

Properties of Exponents

• Product of Powers: Multiply bases with the same exponent: ( a^m \cdot a^n = a^{m+n} ).
• Quotient of Powers: Subtract exponents when dividing same bases: ( a^m / a^n = a^{m-n} ).
• Power of a Power: Multiply exponents: ( (a^m)^n = a^{m \cdot n} ).
• Power of a Product: Distribute exponent over multiplication: ( (ab)^n = a^n \cdot b^n ).
• Power of a Quotient: Distribute exponent over division: ( (a/b)^n = a^n / b^n ).
• Zero Exponent: Any non-zero base raised to the power of zero equals one: ( a^0 = 1 ).
• Negative Exponent: Inverse of the base raised to a positive power: ( a^{-n} = 1/a^n ).

Logarithmic Identities

• Definition: Establishes the inverse relationship between exponentiation and logarithms: if ( b^y = x ), then ( \log_b(x) = y ).
• Logarithm of a Product: Sum of logarithms when multiplying: ( \log_b(xy) = \log_b(x) + \log_b(y) ).
• Logarithm of a Quotient: Difference of logarithms when dividing: ( \log_b(x/y) = \log_b(x) - \log_b(y) ).
• Logarithm of a Power: Coefficient in front when exponentiating: ( \log_b(x^n) = n \cdot \log_b(x) ).
• Change of Base Formula: Converts one logarithmic base to another: ( \log_b(a) = \frac{\log_k(a)}{\log_k(b)} ).
• Logarithm of 1: Always equals zero: ( \log_b(1) = 0 ).
• Logarithm of the Base: Always equals one: ( \log_b(b) = 1 ).

Exponential Growth and Decay

• Exponential Growth Model: Describes growth over time: ( P(t) = P_0 \cdot e^{rt} ), where ( P_0 ) is the initial amount, ( r ) is the growth rate, and ( t ) represents time.
• Exponential Decay Model: Describes a decrease: ( P(t) = P_0 \cdot e^{-rt} ), characterized by a consistent percentage decrease.
• Half-Life: Refers to the time needed for a quantity to reduce to half of its original amount, particularly important in radioactive decay.

Applications of Logarithms

• Solving Exponential Equations: Employ logarithmic identities to isolate the variable effectively.
• pH Scale: Measures hydrogen ion concentration in acidity, defined as ( \text{pH} = -\log_{10}[\text{H}^+] ).
• Richter Scale: Utilizes a logarithmic scale to measure the intensity of earthquakes.
• Sound Intensity: Expressed in decibels, calculated using: ( dB = 10 \log_{10}(I/I_0) ), where ( I_0 ) is a reference intensity.
• Finance: Logarithmic functions facilitate calculations involving compound interest, particularly in modeling growth over time.