Download 8-5 Exponential and Logarithmic Equations

8-5 Exponential and Logarithmic
Equations
Hubarth
Algebra II
An equation of the form 𝑏𝑐𝑥 = 𝑎, where the exponent includes a variable, is an exponential
equation
Ex. 1 Solving an Exponential Equation
Solve 52x = 16.
52x = 16
log 52x = log 16
Take the common logarithm of each side.
2x 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: 52x
52(0.8614)
16
16
Ex. 2 Solving an Exponential Equation by Graphing
Solve 43x = 1100 by graphing.
Graph the equations y1 = 43x and y2 = 1100.
Find the point of intersection.
The solution is x
1.684
Ex. 3 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.
Graph
Property
Change of Base Formula
For any positive numbers , 𝑀, 𝑏, and 𝑐, with 𝑏 ≠ 1 and 𝑐 ≠ 1.
𝑙𝑜𝑔𝑐 𝑀
𝑙𝑜𝑔𝑏 𝑀 =
𝑙𝑜𝑔𝑐 𝑏
Ex. 4 Using the Change of Base Formula
Ex. 4 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
Use the Change of Base Formula.
≈ 0.7782 ≈ 1.387
Use a calculator
log6 12 = log3 x
Write an equation.
1.387 ≈ 𝑙𝑜𝑔3 𝑥
Substitute log6 12 = 1.3868
log 𝑥
1.387 ≈ log 3
Use the change of base formula
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.
𝑥 ≈ 100.6617
Write in exponential form.
≈ 4.589
Use a calculator.
The expression log6 12 is approximately
equal to 1.3869, or log3 4.589.
Ex. 5 Solving a Logarithmic Equation
Solve log (2x – 2) = 4.
Method 1: log (2x – 2) = 4
2x – 2 = 104
Write in exponential form.
2x – 2 = 10000
x = 5001
Solve for x.
Method 2: Graph the equation y1 = log (2x – 2)
and y2 = 4. Use Xmin = 4000, Xmax = 6000,
Ymin = 3.9 and Ymax = 4.1.
Find the point of intersection.
The solution is x = 5001.
Ex. 6 Using Logarithmic Properties to Solve an Equation
Solve 3 log x – log 2 = 5.
3 log x – log 2 = 5
3
Log ( x ) = 5
Write as a single logarithm.
x3
= 105
2
Write in exponential form.
x3 = 2(100,000)
Multiply each side by 2.
2
x = 10
3
200, or about 58.48.
Practice
1. Solve each equation. Round your answer to the nearest ten-thousandth.
2𝑥
𝑥+4
a. 3𝑥 = 4
b.
6
=
21
c.
3
= 101
1.2619
0.8496
2. Solve 116𝑥 = 786 by Graphing
Graph
0.4634
3. Evaluate 𝑙𝑜𝑔5 400 and convert it to a logarithm in base 8
3.7227, 𝑙𝑜𝑔8 2301
4. Solve log 7 − 2𝑥 = −1 . Check your answer.
3.45
5. Solve log 6 − 𝑙𝑜𝑔3𝑥 = −2
200
0.2009