Solving Exponential & Logarithmic Equations 0 ln ln2 3 ln2 3
Solving Exponential & Logarithmic Equations
Solving Exponential & Logarithmic Equations
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<strong>Solving</strong> <strong>Exponential</strong> & <strong>Logarithmic</strong> <strong>Equations</strong> Properties of <strong>Exponential</strong> and <strong>Logarithmic</strong> <strong>Equations</strong>Let be a positive real number such that 1, and let and be real numbers. Then the following properties aretrue:1. if and only if 2. log log if and only if 0, 0 Inverse Properties of Exponents and LogarithmsBase aNatural Base e1. log <strong>ln</strong> 2. <strong>Solving</strong> <strong>Exponential</strong> and <strong>Logarithmic</strong> <strong>Equations</strong>1. To solve an exponential equation, first isolate the exponential expression, then take the logarithm of both sidesof the equation and solve for the variable.2. To solve a logarithmic equation, first isolate the logarithmic expression, then exponentiate both sides of theequation and solve for the variable.For Instance: If you wish to solve the equation, <strong>ln</strong> 2, you exponentiate both sides of the equation to solve itas follows:<strong>ln</strong> 2Original equation Exponentiate both sides Inverse propertyOr you can simply rewrite the logarithmic equation in exponential form to solve (i.e. <strong>ln</strong> 2 if and only if ).Note: You should always check your solution in the original equation.Example 1:Solve each equation.a. 4 64 b. <strong>ln</strong>2 3 <strong>ln</strong> 11Solution:a. 4 64 Original Equationb. <strong>ln</strong>2 3 <strong>ln</strong> 11 Original Equation4 4 Rewrite with like bases2 3 11 Property of logarithmic equations 2 3 Property of exponential equations2 14 Add 3 to both sides 1 Subtract 2 from both sides 7 Divide both sides by 2The solution is 1. Check this in the original equation.The solution is 7. Check this in the original equation.Example 2:Solve 5 20.Solution:5 20 Original Equation 15 Subtract 5 from both sides<strong>ln</strong> <strong>ln</strong> 15 Take the logarithm of both sides 1 <strong>ln</strong> 15 Inverse Property 1 <strong>ln</strong> 15 1.708 Subtract 1 from both sidesCheck:5 20 Original Equation5 . ? 20 Substitute 1.708 for 5 . ? 20 Simplify5 14.999 20 Solution checks
Example 3:Solve the exponential equations.a. 2 7 b. 4 9 c. 2 10Solutions:Method 1: Method 2:a. 2 7 Original Equationa. 2 7 Original Equationlog 2 log 7 Take the logarithm of both sideslog 2 log 7 Take the logarithm of both sideslog 2 log 7 Property of Logarithms log 7 Inverse Property 2.807 Solve for 2.807 Change of Base Formula Method 1: Method 2:b. 4 9 Original Equationb. 4 9 Original Equation 3 3 4.585 Solve for log 4 log 9 Take the logarithm of both sideslog 4 log 9 Take the logarithm of both sides 3 log 4 log 9 Property of Logarithms 3 log 9 Inverse Property 3 Divide both sides by log 4 3 Change of Base Formulac. 2 10 Original Equation 5 Divide both sides by 2<strong>ln</strong> <strong>ln</strong> 5 Take the logarithm of both sides <strong>ln</strong> 5 1.609 Inverse PropertyExample 4:Solve 2 log 5.Solution:2 log 5 Original Equationlog Divide both sides by 24 ⁄ Change to exponential form 32Example 6:Solve 20 <strong>ln</strong> 0.2 30.Solution:Simplify20 <strong>ln</strong> 0.2 30 Original Equation<strong>ln</strong> 0.2 1.5 Divide both sides by 200.2 . Change to exponential form 5 . 22.408 Divide both sides by 0.2Example 5:Solve 3 log 6.Solution:3 log 6 Original Equationlog 2 Divide both sides by 310 Change to exponential form 100SimplifyExample 7: <strong>Solving</strong> a <strong>Logarithmic</strong> Equationusing ExponentiationSolve log 2 log 3 1Solution:log 2 log 3 1 Original Equationlog 1 Condense the left side3 3 Exponentiate both sides 3Inverse Property2 3 9 Multiply both sides by 3 9Solve for
Practice ProblemsSolve the following equations:Remember that the arguments of all logarithms must be greater than 0. Also exponentials in the form of will be greater than 0. Be sure to check all your answers in the original equation.1. 3 812. 8 43. 54. 14 3 115. 6 <strong>ln</strong> 3 06. log3 1 27. <strong>ln</strong> <strong>ln</strong> 3 48. 2 <strong>ln</strong> 3 49. 5 410. <strong>ln</strong> 2 611. 4 0.2512. 2 5 3 013. log 3 log log 3214. 2 log 4 015. log log 3 216. log 5 log 2 317. 4 <strong>ln</strong>2 3 1118. log log 6 2 log 419. 2 6420. 5 2521. 4 22. 3 8123. log 524. log 325. log 2 log 10026. <strong>ln</strong> 4 <strong>ln</strong> 727. log 2 1 228. log 10 229. 3 50030. 8 100031. <strong>ln</strong> 7.2532. <strong>ln</strong> 0.533. 2 . 4534. 100 . 2035. 121 4 1836. 251 1237. log 2 1.538. log 2 0.6539. log 5 740. 4 log 1 4.841. log log 3 342. 2 log log 1 1
Practice Problems Answers1. 52. 3. 1.6094. 2.1205. 134.4766. 337. 163.7948. 2.4639. -1.13910. 18.086, -22.08611. 12. 1.09913. 14. 15. 416. 317. 6.32118. 9619. 620. 221. 122. 623. 24324. 6425. 5026. 327. 428. 3529. 5.6630. 3.3231. 1408.1032. 0.6133. 6.2334. 2.6835. No Solution36. –0.6537. 15.8138. 0.3239. 6440. 5.9041. 42. 2
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