Exponents & Logarithms formulas

All 20 formulas in this topic

Product Rule for Exponentsam⋅an=am+na^m \cdot a^n = a^{m+n}The Product Rule for Exponents simplifies multiplication of powers with the same base by adding the exponents into a single power.Quotient Rule for Exponentsaman=am−n\frac{a^m}{a^n} = a^{m-n}Quotient Rule for Exponents: when dividing powers with the same base, subtract the exponents to simplify the expression.Power Rule for Exponents(am)n=amn(a^m)^n = a^{mn}Simplify nested exponents by multiplying the exponents together: this is the fundamental tool you need whenever one exponent is itself raised to a power.Zero Exponenta0=1,a≠0a^0 = 1, \quad a \neq 0Any non-zero number raised to the power of zero always equals one, providing a key rule for simplifying exponential expressions and solving equations.Negative Exponenta−n=1ana^{-n} = \frac{1}{a^n}A negative exponent tells you to take the reciprocal: a^{-n} equals 1/a^n. Use this to rewrite expressions with negative powers.Fractional Exponentam/n=amna^{m/n} = \sqrt[n]{a^m}Convert between radicals and exponents: a fractional exponent means the denominator is the root and the numerator is the power on the inside.Simplifying Radicalsab=a b,a,b≥0\sqrt{ab} = \sqrt{a}\,\sqrt{b}, \quad a, b \ge 0Break down square roots of products into simpler parts you can work with—essential for simplifying radical expressions in algebra.Rationalizing the Denominator1a=aa\frac{1}{\sqrt{a}} = \frac{\sqrt{a}}{a}Rationalizing the Denominator removes square roots from the bottom of a fraction, making it easier to compare values and perform calculations.Definition of a Logarithmlog⁡ax=y  ⟺  ay=x\log_a x = y \iff a^y = xUse the definition of a logarithm to convert between exponential and logarithmic form when solving for unknown exponents.Product Rule for Logarithmslog⁡a(xy)=log⁡ax+log⁡ay\log_a(xy) = \log_a x + \log_a yCombine multiplication inside a logarithm into a sum of separate logarithms, making complex equations and expressions simpler to solve and evaluate.Quotient Rule for Logarithmslog⁡a ⁣(xy)=log⁡ax−log⁡ay\log_a\!\left(\frac{x}{y}\right) = \log_a x - \log_a ySimplifies logarithms of fractions by converting division into subtraction, letting you break complex logs into simpler pieces.Power Rule for Logarithmslog⁡a(xn)=nlog⁡ax\log_a(x^n) = n \log_a xThe Power Rule for Logarithms lets you pull an exponent out of a logarithm as a multiplier, making complex logs easier to solve and simplify.Change of Base Formulalog⁡ax=log⁡bxlog⁡ba\log_a x = \frac{\log_b x}{\log_b a}Convert logarithms between different bases so you can evaluate them on a calculator or combine logarithms with different bases.Natural Logarithmln⁡x=log⁡ex\ln x = \log_e xNatural logarithm finds the power to which e must be raised to get a number, and it's essential for solving equations involving exponential functions.Exponential GrowthA=A0ekt,k>0A = A_0 e^{kt}, \quad k > 0Use Exponential Growth when a quantity grows by multiplying by a constant factor each time period, like bacteria populations or money earning compound interest.Exponential DecayA=A0e−kt,k>0A = A_0 e^{-kt}, \quad k > 0Exponential Decay models quantities that shrink at a rate proportional to their current amount, used for radioactive decay, cooling, and depreciation.Half-Life FormulaA=A0(12)t/t1/2A = A_0 \left(\tfrac{1}{2}\right)^{t / t_{1/2}}The Half-Life Formula determines how much of a radioactive or decaying substance remains after a given time period has passed.Doubling Timet2=ln⁡2kt_2 = \frac{\ln 2}{k}Doubling Time calculates how long it takes for a quantity to double in exponential growth when you know the continuous growth rate.Continuous Compound InterestA=PertA = Pe^{rt}Calculate how much money grows continuously over time, or solve how long growth takes at a given rate of compound interest.Euler's Numbere=lim⁡n→∞(1+1n)ne = \lim_{n \to \infty} \left(1 + \frac{1}{n}\right)^{n}Euler's Number e is a fundamental constant approximately equal to 2.71828 that models continuous exponential growth and decay.

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