* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Download logarithmic equation
Survey
Document related concepts
Mathematical optimization wikipedia , lookup
Lateral computing wikipedia , lookup
Computational complexity theory wikipedia , lookup
Genetic algorithm wikipedia , lookup
Computational fluid dynamics wikipedia , lookup
Routhian mechanics wikipedia , lookup
Inverse problem wikipedia , lookup
Simplex algorithm wikipedia , lookup
Travelling salesman problem wikipedia , lookup
Perturbation theory wikipedia , lookup
Signal-flow graph wikipedia , lookup
Computational electromagnetics wikipedia , lookup
System of polynomial equations wikipedia , lookup
Transcript
Warm Up Solve. 1. log16x = 3 2 2. logx8 = 3 3. log10,000 = x 64 1.1 4 7-4 Logarithmic Equations and Inequalities TEXTBOOK PAGE 478 An exponential equation is an equation containing one or more expressions that have a variable as an exponent. To solve exponential equations: • Try writing them so that the bases are all the same. • Take the logarithm of both sides. A logarithmic equation is an equation with a logarithmic expression that contains a variable. You can solve logarithmic equations by using the properties of logarithms. Solving by Rewriting as an Exponential Solve log4(x+3) = 2 42 = x+3 16 = x+3 13 = x Example 1 Solve for x: log 6 x 2 Solution: Let’s rewrite the problem in exponential form. 6 x 2 We’re finished ! Example 2 1 Solve for y: log 5 y 25 Solution: Rewrite the problem in exponential form. 1 5 25 y 2 5 5 y y 2 1 Since 5 2 25 Example 3 Evaluate log3 27. Solution: Try setting this up like this: log3 27 y Now rewrite in exponential form. 3 27 y 3 3 y y3 3 Example 4 2 Evaluate: log7 7 Solution: log7 7 y 2 7 7 y2 y 2 First, we write the problem with a variable. Now take it out of the logarithmic form and write it in exponential form. Example 5 Evaluate: 4 log 4 16 Solution: 4 log 4 16 y First, we write the problem with a variable. log4 y log4 16 Now take it out of the exponential form and write it in logarithmic form. Just like 2 8 converts to log2 8 3 3 y 16 Finally, we want to take a look at the Property of Equality for Logarithmic Functions. Suppose b 0 and b 1. Then logb x1 log b x 2 if and only if x1 x 2 Basically, with logarithmic functions, if the bases match on both sides of the equal sign , then simply set the arguments equal. Example 1 Solve: log3 (4x 10) log3 (x 1) Solution: Since the bases are both ‘3’ we simply set the arguments equal. 4x 10 x 1 3x 10 1 3x 9 x 3 Caution Watch out for calculated solutions that are not solutions of the original equation. Example 2 Solve: log8 (x 14) log8 (5x) 2 Solution: Since the bases are both ‘8’ we simply set the arguments equal. 2 x 14 5x 2 x 5x 14 0 (x 7)(x 2) 0 Factor (x 7) 0 or (x 2) 0 x 7 or x 2 continued on the next page Example 2 continued Solve: log8 (x 14) log8 (5x) 2 Solution: x 7 or x 2 It appears that we have 2 solutions here. If we take a closer look at the definition of a logarithm however, we will see that not only must we use positive bases, but also we see that the arguments must be positive as well. Therefore -2 is not a solution. Let’s end this lesson by taking a closer look at this. Use a table and graph to solve 2x = 4x – 1. Use a graphing calculator. Enter 2x as Y1 and 4(x – 1) as Y2. In the table, find the x-values where Y1 is equal to Y2. The solution is x = 2. In the graph, find the x-value at the point of intersection. Use a table and graph to solve 2x > 4x – 1. Use a graphing calculator. Enter 2x as Y1 and 4(x – 1) as Y2. In the table, find the x-values where Y1 is greater than Y2. The solution is x < 2. In the graph, find the x-value at the point of intersection. Use a table and graph to solve log x2 = 6. Use a graphing calculator. Enter log(x2) as Y1 and 6 as Y2. In the table, find the x-values where Y1 is equal to Y2. The solution is x = 1000. In the graph, find the x-value at the point of intersection.