Exponential and Logarithmic Equations and Models College Algebra
Product Rule for Logarithms The product rule for logarithms can be used to simplify a logarithm of a product by rewriting it as a sum of individual logarithms. log $ (MN) = log $ M + log $ N for b > 0 Example: Expand log. 30x 3x + 4 Solution: = log. 30x + log. 30x + 4 = log. 30 + log. x + log. 30x + 4
Quotient Rule for Logarithms The quotient rule for logarithms can be used to simplify a logarithm of a quotient by rewriting it as the difference of individual logarithms. M log $ N = log $ M log $ N Example: Expand log 345 674.468 Solution: = log 34(46.).(46.) = log 34. = log 2 + log x log 3
Power Rule for Logarithms The power rule for logarithms can be used to simplify a logarithm of a power by rewriting it as the product of the exponent times the logarithm of the base. log $ M : = nlog $ M Example: Expand log. 25 Solution: = log. 5 3 = 2 log. 5
Expand Logarithmic Expressions Using the Logarithm Rules Taken together, the product rule, quotient rule, and power rule are often called laws of logs. Sometimes we apply more than one rule in order to simplify an expression. Example: Expand log 3 4 5 Solution: = > = log 3 x 3 log 3 y. = 2 log 3 x 3 log 3 y
Condense Logarithmic Expressions Given a sum, difference, or product of logarithms with the same base, write an equivalent expression as a single logarithm: 1. Apply the power property first. Identify terms that are products of factors and a logarithm, and rewrite each as the logarithm of a power. 2. Next apply the product property. Rewrite sums of logarithms as the logarithm of a product. 3. Apply the quotient property last. Rewrite differences of logarithms as the logarithm of a quotient.
Change of Base for Logarithms The change-of-base formula can be used to evaluate a logarithm with any base. For any positive real numbers M, b, and n, where n 1 and b 1, log $ M = log : M log : b It follows that the change-of-base formula can be used to rewrite a logarithm with any base as the quotient of common or natural logs. ln M log M log $ M = = ln b log b
Desmos Interactive Topic: change of base https://www.desmos.com/calculator/umnz24xgl1
Exponential Equations The one-to-one property of exponential functions can be used to solve exponential equations. For any algebraic expressions S and T, and any positive real number b 1, b E = b F if and only if S = T Given an exponential equation with the form b S = b T, where S and T are algebraic expressions with an unknown, solve for the unknown: 1. Use the rules of exponents to simplify, if necessary, so that the resulting equation has the form b J = b F. 2. Use the one-to-one property to set the exponents equal. 3. Solve the resulting equation, S = T, for the unknown.
Exponential Equations with a Common Base Solve: 2 4KL = 2 34KM Solution: 2 4KL = 2 34KM The common base is 2 x 1 = 2x 4 Use the one-to-one property x = 3 Solve for x
Exponential Equations with Unlike Bases Solve: 8 463 = 16 46L Solution: 8 463 = 16 46L 2. 463 = 2 M 46L Write 8 and 16 as powers of 2 2.467 = 2 M46M To take a power of a power, multiply exponents 3x + 6 = 4x + 4 x = 2 Use the one-to-one property to set the exponents equal Solve for x
Use Logarithms to Solve Exponential Equations Given an exponential equation in which a common base cannot be found, solve for the unknown. 1. Apply the logarithm of both sides of the equation. Use the natural logarithm unless one of the terms in the equation has base 10. 2. Use the rules of logarithms to solve for the unknown. Example: Solve 5 463 = 4 4 ln 5 463 = ln 4 4 x + 2 ln 5 = x ln 4 x ln 5 ln 4 = 2 ln 5 Use the natural logarithm on both sides Use the laws of logs Rearrange x ln Q M = ln L 3Q Use the laws of logs, then solve x = RS T.TM RS L.3Q
Equations Containing e One common type of exponential equations are those with base e. When we have an equation with a base e on either side, we can use the natural logarithm to solve it. Given an equation of the form y = Ae kt, solve for t. 1. Divide both sides of the equation by A. 2. Apply the natural logarithm of both sides of the equation. 3. Divide both sides of the equation by k.
Extraneous Solutions An extraneous solution is a solution that is correct algebraically but does not satisfy the conditions of the original equation. When the logarithm is taken on both sides of the equation, remember that the argument of the logarithm must be positive. If the number we are evaluating in a logarithm function is negative, there is no output. Example: e 34 e 4 56 = 0 e 4 + 7 e 4 8 = 0 Factor the quadratic equation e 4 = 7 or e 4 = 8 x = ln 8 Find the zeros Since x = ln 7 is not a real number, it is an extraneous solution
Logarithmic Equations Use the definition of a logarithm to solve logarithmic equations. For any algebraic expression S and real numbers b and c where b > 0, b 1 log $ S = c if and only if b` = S Example: log 3 2 + log 3 3x 5 = 3 log 3 6x 10 = 3 Apply the product rule 6x 10 = 2. Apply the definition of a logarithm x = 3 Solve for x
Logarithmic Equations Use the one-to-one property of logarithms to solve logarithmic equations. For any real numbers x > 0, S > 0, T > 0 and any positive real number b where b 1, log $ S = log $ T if and only if S = T Example: log 3x 2 log 2 = log x + 4 log.4k3 3.4K3 3 x = 10 = log x + 4 Apply the quotient rule = x + 4 Apply the one-to-one property Solve for x
Exponential Growth The exponential growth function can be used to model the real-world phenomenon of rapid growth. y = A T e de where A T is equal to the value at time zero and k is a positive constant that determines the rate of growth. This function can be used in applications involving the doubling time, the time it takes for a quantity to double from it s initial value: 2A T = A T e de ln 2 = kt t = ln 2 k
Exponential Decay The exponential decay model is used when the quantity falls rapidly toward zero. y = A T e Kde where A T is equal to the value at time zero and k is a negative constant that determines the rate of decay. The half-life is the time it takes for a substance to exponentially decay to half of its original quantity. 1 2 A T = A T e de ln L 3 = kt, or ln 2 = kt t = ln 2 k
Desmos Interactive Topic: exponential growth and decay https://www.desmos.com/calculator/4zae7vnjfr
Quick Review What are the three rules that comprise the laws of logs? Can we expand ln(x 3 + y 3 )? Can we change common logarithms to natural logarithms? What is the one-to-one property for exponential functions? What is an extraneous solution? What is the half-life of an exponential decay model? Is there any way to solve 2 4 = 3 4? How can we solve log 3 7 on a calculator that has ln and log buttons?