8 5 Exponential and Logarithmic Equations Solving logarithmic

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 52 x = 16 52 x = log 16 Take the common logarithm of each side. 2 x log 5 = log 16 Use the power property of logarithms. x= log 16 2 log 5 Divide each side by 2 log 5. 0. 8614 Use a calculator. Check: 52 x 52(0. 8614) 16 16

Solving an Exponential Equation by Graphing Solve 43 x = 1100 by graphing. Graph the equations y = 43 x 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 log 6 12 is approximately equal to 1. 3869, or log 3 4. 589.

Solving an Exponential Equation by Tables Solve 52 x = 120 using tables. Enter y 1 = 52 x – 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, logc. M logb. M = logcb

Using the Change of Base Formula Use the Change of Base Formula to evaluate log 6 12. Then convert log 6 12 to a logarithm in base 3. log 6 12 = log 12 log 6 1. 0792 0. 7782 log 6 12 = log 3 x Use the Change of Base Formula. 1. 387 Use a calculator. Write an equation. 1. 387 log 3 x Substitute log 6 12 = 1. 3868 1. 387 log x log 3 Use the Change of Base Formula.

Solving a Logarithmic Equation Solve log (2 x – 2) = 4 2 x – 2 = 104 Write in exponential form. 2 x – 2 = 10000 x = 5001 Solve for x. Check: log (2 x – 2) 4 log (2 • 5001 – 2) 4 log 10, 000 4 log 104 = 4

Using Logarithmic Properties to Solve an Equation Solve 3 log x – log 2 = 5 Log ( x ) = 5 2 x 3 = 105 2 3 Write as a single logarithm. Write in exponential form. x 3 = 2(100, 000) 3 x = 10 The solution is 10 3 Multiply each side by 2. 200, or about 58. 48.

Homework 8 -5 Pg 464 # 1, 2, 13, 14, 23, 25, 26, 33, 34, 42, 43