8-5 Exponential and Logarithmic Equations Solving logarithmic & exponential equations Objectives Solving Exponential Equations Solving Logarithmic Equations Vocabulary An equation of the form bcx = a is and exponential equation. If m = n, then log m = log n Solving an Exponential Equation Solve log 52x = 16.
52x = 16 52x = log 16 Take the common logarithm of each side. 2x log 5 = log 16 Use the power property of logarithms. log 16 x= 2 log 5 Divide each side by 2 log 5. 0.8614 Check: 52x
52(0.8614) 16 16 Use a calculator. Solving an Exponential Equation by Graphing Solve 43x = 1100 by graphing. Graph the equations y = 43x and y = 1100. Find the point of intersection. The solution is x 1.684
Continued (continued) 1.387 log 3 log x Multiply each side by log 3. 1.387 0.4771 log x Use a calculator. 0.6617 log x
Simplify. 100.6617 Write in exponential form. 4.589 Use a calculator. x The expression log6 12 is approximately equal to 1.3869, or log3 4.589. Solving an Exponential Equation by Tables
Solve 52x = 120 using tables. Enter y1 = 52x 120. Use tabular zoom-in to find the sign change, as shown at the right. The solution is x 1.487. Real-World Example The population of trout in a certain stretch of the Platte River is shown for five consecutive years in the table, where 0 represents the year 1997. If the decay rate remains constant, in the beginning of which year might at most 100 trout remain in this stretch of river? Time t Pop. P(t) 0
1 2 3 4 5000 4000 3201 2561 2049
Step 1: Enter the data into your calculator. Step 2: Use the Exp Reg feature to find the exponential function that fits the data. Continued (continued) Step 3: Graph the function and the line y = 100. Step 4: Find the point of intersection. The solution is x 18, so there may be only 100 trout remaining in the beginning of the year 2015. Vocabulary For any positive numbers, M, b, c, with b 1 and c 1,
logcM logbM = logcb Using the Change of Base Formula Use the Change of Base Formula to evaluate log6 12. Then convert log6 12 to a logarithm in base 3. log6 12 = log 12 log 6 1.0792 0.7782 log6 12 = log3 x Use the Change of Base Formula. 1.387
Use a calculator. Write an equation. 1.387 log3x Substitute log6 12 = 1.3868 1.387 log x log 3 Use the Change of Base Formula. Solving a Logarithmic Equation
4 log 10,000 4 log 104 = 4 Using Logarithmic Properties to Solve an Equation Solve 3 log x log 2 = 5. 3 log x log 2 = 5 Log ( x ) = 5 2 x3 = 105 2 3
Write as a single logarithm. Write in exponential form. x3 = 2(100,000) 3 x = 10 The solution is 10 3 Multiply each side by 2. 200, or about 58.48. 200, or about 58.48.
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