Exponential Equations An exponential equation is one in which the variable occurs in the exponent. For example, 2x = 7 The variable x presents a difficulty because it is in the exponent. To deal with this difficulty, we take the logarithm of each side and then use the Laws of Logarithms to “bring down x” from the exponent.

Exponential Equations ln 2x = ln 7 x ln 2 = ln 7  2.807 Given Equation Take In of each side Law 3 (Bring down exponent) Solve for x Calculator

Exponential Equations Recall that Law 3 of the Laws of Logarithms says that loga AC = C loga A. The method that we used to solve 2x = 7 is typical of how we solve exponential equations in general.

Example 1 – Solving an Exponential Equation Find the solution of the equation 3x + 2 = 7, rounded to six decimal places. Solution: We take the common logarithm of each side and use Law 3. 3x + 2 = 7 log(3x + 2) = log 7 Given Equation Take log of each side

Example 1 – Solution (x + 2)log 3 = log 7 x + 2 =  –0.228756 cont’d Law 3 (bring down exponent) Divide by log 3 Subtract 2 Calculator

Example 1 – Solution Check Your Answer cont’d Check Your Answer Substituting x = –0.228756 into the original equation and using a calculator, we get 3(–0.228756) + 2  7

Example 4 – An Exponential Equation of Quadratic Type Solve the equation e2x – ex – 6 = 0. Solution: To isolate the exponential term, we factor. e2x – ex – 6 = 0 (ex)2 – ex – 6 = 0 Given Equation Law of Exponents

Example 4 – Solution (ex – 3)(ex + 2) = 0 ex – 3 = 0 or ex + 2 = 0 cont’d (ex – 3)(ex + 2) = 0 ex – 3 = 0 or ex + 2 = 0 ex = 3 ex = –2 The equation ex = 3 leads to x = ln 3. But the equation ex = –2 has no solution because ex > 0 for all x. Thus, x = ln 3  1.0986 is the only solution. Factor (a quadratic in ex) Zero-Product Property

Logarithmic Equations

Logarithmic Equations A logarithmic equation is one in which a logarithm of the variable occurs. For example, log2(x + 2) = 5 To solve for x, we write the equation in exponential form. x + 2 = 25 x = 32 – 2 = 30 Exponential form Solve for x

Logarithmic Equations Another way of looking at the first step is to raise the base, 2, to each side of the equation. 2log2(x + 2) = 25 x + 2 = 25 x = 32 – 2 = 30 Raise 2 to each side Property of logarithms Solve for x

Logarithmic Equations The method used to solve this simple problem is typical. We summarize the steps as follows.

Example 6 – Solving Logarithmic Equations Solve each equation for x. (a) ln x = 8 (b) log2(25 – x) = 3 Solution: (a) ln x = 8 x = e8 Therefore, x = e8  2981. Given equation Exponential form

Example 6 – Solution We can also solve this problem another way: cont’d We can also solve this problem another way: ln x = 8 eln x = e8 x = e8 Given equation Raise e to each side Property of ln

Example 6 – Solution cont’d (b) The first step is to rewrite the equation in exponential form. log2(25 – x) = 3 25 – x = 23 25 – x = 8 x = 25 – 8 = 17 Given equation Exponential form (or raise 2 to each side)

Example 6 – Solution Check Your Answer If x = 17, we get cont’d Check Your Answer If x = 17, we get log2(25 – 17) = log2 8 = 3