Exponential and Logarithmic Functions Cheat Sheet

The core ideas of Exponential and Logarithmic Functions distilled into a single, scannable reference — perfect for review or quick lookup.

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Quick Reference

Exponential Function

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.

Logarithm

The inverse of exponentiation. log_b(x) = y means b^y = x. It answers: what exponent of base b gives x?

Natural Logarithm (ln)

The logarithm with base e (approximately 2.71828). Written as ln(x) = log_e(x). Central to calculus and continuous growth models.

Product Rule of Logarithms

log_b(MN) = log_b(M) + log_b(N). The logarithm of a product equals the sum of the logarithms.

Quotient Rule of Logarithms

log_b(M/N) = log_b(M) - log_b(N). The logarithm of a quotient equals the difference of the logarithms.

Power Rule of Logarithms

log_b(M^n) = n * log_b(M). The logarithm of a power equals the exponent times the logarithm of the base.

Change of Base Formula

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.

Exponential Growth and Decay

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.

Compound Interest Formula

A = P(1 + r/n)^(nt) where P is principal, r is annual rate, n is compounding frequency, and t is time in years. Continuous compounding: A = Pe^(rt).

Asymptote of Exponential/Log Functions

Exponential functions have a horizontal asymptote (typically y = 0). Logarithmic functions have a vertical asymptote (typically x = 0). The function approaches but never reaches the asymptote.

Key Terms at a Glance

Exponential Function:A function of the form f(x) = a * b^x where b > 0 and b is not equal to 1.

Logarithm:The inverse of exponentiation. log_b(x) = y means b^y = x.

Natural Logarithm (ln):Logarithm with base e, approximately 2.71828. Written as ln(x).

Common Logarithm (log):Logarithm with base 10. Often written as log(x) without a subscript.

Exponential Growth:Growth where the rate of increase is proportional to the current value, modeled by f(x) = a * b^x with b > 1.

Exponential Decay:Decrease where the rate of decline is proportional to the current value, modeled by f(x) = a * b^x with 0 < b < 1.

Euler's Number (e):An irrational mathematical constant approximately equal to 2.71828, the base of the natural logarithm.

Asymptote:A line that a curve approaches but never reaches. Exponential functions have horizontal asymptotes; logarithmic functions have vertical asymptotes.

Half-Life:The time required for a quantity undergoing exponential decay to reduce to half its initial value.

Compound Interest:Interest calculated on both the initial principal and previously earned interest: A = P(1 + r/n)^(nt).

Continuous Compounding:Interest compounded infinitely often, modeled by A = Pe^(rt).

Change of Base Formula:log_b(x) = log_c(x) / log_c(b), used to evaluate logarithms with non-standard bases using a calculator.

Domain:The set of all valid input values for a function. For logarithmic functions, the argument must be positive.

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