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Session Length
~15 min
Adaptive Checks
14 questions
Transfer Probes
9
Exponential functions model quantities that grow or decay at a rate proportional to their current value. The general form f(x) = a * b^x describes exponential growth when b > 1 and exponential decay when 0 < b < 1. The natural exponential function f(x) = e^x, where e is approximately 2.71828, is fundamental in calculus, finance, and the natural sciences.
Logarithmic functions are the inverses of exponential functions. The logarithm log_b(x) answers the question: to what power must the base b be raised to produce x? Common bases include 10 (common log), e (natural log, ln), and 2 (used in computer science). Properties of logarithms -- the product rule, quotient rule, power rule, and change of base formula -- are essential tools for simplifying expressions and solving exponential equations.
These functions appear throughout real-world applications: compound interest and continuous compounding in finance, population growth and radioactive decay in science, pH calculations in chemistry, the Richter scale for earthquakes, and decibel scales for sound intensity. Solving exponential and logarithmic equations requires fluency with logarithm properties and the ability to convert between exponential and logarithmic forms. This topic is central to AP Precalculus and serves as a gateway to calculus concepts like the derivative of e^x.
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A function of the form f(x) = a * b^x where a is the initial value and b is the base. Exhibits growth when b > 1 and decay when 0 < b < 1.
Example: f(x) = 100 * (1.05)^x models an investment growing at 5% per year, where 100 is the initial amount.
The inverse of exponentiation. log_b(x) = y means b^y = x. It answers: what exponent of base b gives x?
Example: log_2(8) = 3 because 2^3 = 8. log_10(1000) = 3 because 10^3 = 1000.
The logarithm with base e (approximately 2.71828). Written as ln(x) = log_e(x). Central to calculus and continuous growth models.
Example: ln(e) = 1, ln(e^3) = 3, ln(1) = 0.
log_b(MN) = log_b(M) + log_b(N). The logarithm of a product equals the sum of the logarithms.
Example: log(6) = log(2 * 3) = log(2) + log(3).
log_b(M/N) = log_b(M) - log_b(N). The logarithm of a quotient equals the difference of the logarithms.
Example: log(5/2) = log(5) - log(2).
log_b(M^n) = n * log_b(M). The logarithm of a power equals the exponent times the logarithm of the base.
Example: log(x^3) = 3 * log(x). ln(e^5) = 5 * ln(e) = 5.
log_b(x) = log_c(x) / log_c(b) for any valid base c. Allows conversion between logarithm bases, typically to base 10 or e for calculator use.
Example: log_2(10) = log(10) / log(2) = 1 / 0.301 = 3.322.
Growth occurs when the base b > 1 (quantity increases over time). Decay occurs when 0 < b < 1 (quantity decreases). Rate models: A = A_0 * e^(kt) where k > 0 is growth and k < 0 is decay.
Example: Radioactive decay: A(t) = 100 * e^(-0.05t) models 100 grams of a substance decaying with rate constant k = -0.05.
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