Questions and Answers
What is the y-intercept of the exponential function defined by the equation $f(x) = 3 imes 2^x$?
Which of the following represents the correct use of the product of powers property?
Which identity defines a logarithm of a power?
What does the term 'half-life' refer to in exponential decay?
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Which logarithmic identity can be used to isolate the variable in the equation $b^y = x$?
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Study Notes
Graphing Exponential Functions
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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).
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Key Features:
- Y-Intercept: ( (0, a) ).
- Horizontal Asymptote: ( y = 0 ) (as ( x \to -\infty )).
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Increasing/Decreasing:
- Increasing if ( b > 1 ).
- Decreasing if ( 0 < b < 1 ).
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Domain & Range:
- Domain: ( (-\infty, \infty) ).
- Range: ( (0, \infty) ) for ( a > 0 ) or ( (-\infty, 0) ) for ( a < 0 ).
Properties of Exponents
- Product of Powers: ( a^m \cdot a^n = a^{m+n} )
- Quotient of Powers: ( a^m / a^n = a^{m-n} )
- Power of a Power: ( (a^m)^n = a^{m \cdot n} )
- Power of a Product: ( (ab)^n = a^n \cdot b^n )
- Power of a Quotient: ( (a/b)^n = a^n / b^n )
- Zero Exponent: ( a^0 = 1 ) (where ( a \neq 0 ))
- Negative Exponent: ( a^{-n} = 1/a^n )
Logarithmic Identities
- Definition: If ( b^y = x ), then ( \log_b(x) = y ).
- Logarithm of a Product: ( \log_b(xy) = \log_b(x) + \log_b(y) )
- Logarithm of a Quotient: ( \log_b(x/y) = \log_b(x) - \log_b(y) )
- Logarithm of a Power: ( \log_b(x^n) = n \cdot \log_b(x) )
- Change of Base Formula: ( \log_b(a) = \frac{\log_k(a)}{\log_k(b)} )
- Logarithm of 1: ( \log_b(1) = 0 )
- Logarithm of the Base: ( \log_b(b) = 1 )
Exponential Growth and Decay
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Exponential Growth:
- Model: ( P(t) = P_0 \cdot e^{rt} )
- ( P_0 ): initial amount, ( r ): growth rate, ( t ): time.
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Exponential Decay:
- Model: ( P(t) = P_0 \cdot e^{-rt} )
- Characterized by a constant percentage decrease.
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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.
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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 ).
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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.
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Description
This quiz covers the fundamental concepts of graphing exponential functions and the properties of exponents. Learn about the basic form of exponential functions, key features like y-intercepts and asymptotes, as well as important exponent rules. Test your knowledge and understanding of these crucial mathematical concepts.