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Section 3.4 Exponential & Logarithmic Functions Remember our properties . . . Properties of Logarithms For 𝑀 > 0 and 𝑁 > 0: 1. log 𝑏 𝑀𝑁 = log 𝑏 𝑀 + log 𝑏 𝑁 2. 𝑀 log 𝑏 𝑁 = log 𝑏 𝑀 − log 𝑏 𝑁 3. log 𝑏 𝑀𝑝 = 𝑝 ∙ log 𝑀 Product Rule Quotient Rule Power Rule Change of Base Formula • Can be used to compute logarithms that aren’t base 10 or base 𝑒. • Change of Base Formula: log 𝑀 ln 𝑀 log 𝑀 𝑁 = = log 𝑁 ln 𝑁 Exponential Equations • An exponential equation is an equation containing a variable in an exponent. • Ex’s: 23𝑥−8 = 16, 4𝑥 = 15, 40𝑒 0.6𝑥 = 240 • So, how do we solve exponential equations? Example 1 Solve. a. 53𝑥−6 = 125 b. 8𝑥+2 = 4𝑥−3 Summary of Method Used in Ex 1 • In Ex 1, we could rewrite both sides of the equation using the same base. What do we do here? • Our previous method won’t work on these. One Important Thing to Remember • The logarithmic function is not defined for zero or negative numbers. • So, we need to check our solutions! Example 2 Solve. a. 4𝑥 = 15 b. 10𝑥 = 120000 Summary of Method Used in Ex 2 • This method is used when we can’t rewrite each side using the same base. Example 3 Solve. 40𝑒 0.6𝑥 − 3 = 237 Example 4 Solve. 5 𝑥−2 2𝑥+3 =4 Example 5 Solve. 𝑒 2𝑥 𝑥 − 4𝑒 + 3 = 0 Some Factoring Practice a. 𝑒 2𝑥 − 4𝑒 𝑥 + 3 = b. 𝑒 2𝑥 − 9 = c. 4𝑒 2𝑥 − 12𝑒 𝑥 + 9 = What about our logarithms? • A logarithmic equation is an equation containing a variable in a logarithmic expression. • Ex’s: log 4 (𝑥 + 3) = 2 and ln(4𝑥 + 3) = ln • Note: When solving logarithmic equations, we always need to check our solution! 1 𝑥 Example 6 Solve. log 2 (𝑥 − 4) = 3 Check. Example 7 Solve. 4 ln 3𝑥 = 8 Check. Example 8 log 2 𝑥 + log 2 (𝑥 − 7) = 3 Check: Example 9 1 ln 𝑥 + 2 − ln(4𝑥 + 3) = ln 𝑥 Check: Example 10 The formula 𝐴 = 36.1𝑒 0.0126𝑡 models the population of California, 𝐴, in millions, 𝑡 years after 2005. a. What was the population of California in 2005? b. When will the population of California reach 40 million? Example 11 How long will it take $25,000 to grow to $500,000 at 9% annual interest compounded monthly? 𝑟 𝑛𝑡 Don’t forget: 𝐴 = 𝑃 1 + 𝑛 Questions??? Make sure to be working in MyMathLab!!!