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Aim: How do we solve exponential equations using logarithms?. Do Now:. Solving Exponential Equations. What is an exponential equation?. An exponential equation is an equation, in which the variable is an exponent. Ex. 3 x - 4 = 9. - PowerPoint PPT Presentation
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Aim: Exponential Equations using Logs
Course: Alg. 2 & Trig.
Aim: How do we solve exponential equations using logarithms?
Do Now:
Aim: Exponential Equations using Logs
Course: Alg. 2 & Trig.
Solving Exponential Equations
What is an exponential equation?
An exponential equation is an equation, in which the variable is an exponent. Ex. 3x - 4 = 9
Question: What power of 3 will give us 9?Answer: 2 x - 4 = 2 and x = 6
Solution of this equation was possible because 9 is a power of 3 or in general terms
For b > 0, and b 1, bx = by x = y
To solve an exponential equation, write eachside as a power of the same base.
3x - 4 = 32
x - 4 = 2rewrite each side with same base
equate the exponents and solve
x = 6
Aim: Exponential Equations using Logs
Course: Alg. 2 & Trig.
Exponential Equation Problems
Solve and check 5x + 1 = 54
Solve and check 2x - 1 = 82
x + 1 = 4 bx = by x = y5x + 1 = 54
x = 3
Check 53 + 1 = 54 54 = 54
2x - 1 = 82 convert to like bases8 = 23
2x - 1 = (23)2 = 26
x – 1 = 6 bx = by x = yx = 7
Check 27 - 1 = 82 26 = 64 = 82
Aim: Exponential Equations using Logs
Course: Alg. 2 & Trig.
Exponential Equation Problems
Solve and check 9x + 1 = 27x
9x + 1 = 27x
(32)x + 1 = (33)x
2x + 2 = 3x
2 = x
convert to like bases9 = 32; 27 = 33
distributive property;product of powers property32x + 2 = 33x
bx = by x = y
Check 92 + 1 = 272
93 = 272
729 = 729
Aim: Exponential Equations using Logs
Course: Alg. 2 & Trig.
Exponential Equation Problems
Solve and check
(1
4)x 81 x
convert to like bases1/4 = 2-2; 8 = 23
distributive property;product of powers property
bx = by x = y
(2 2)x (23)1 x
(1
4)x 81 x
22x23 3x
-2x = 3 – 3x
x = 3
Check
(1
4)3 81 3 8 2
1
64
1
82 1
64
Aim: Exponential Equations using Logs
Course: Alg. 2 & Trig.
Solving Exponential Equations w/Logs
What if the two sides of the exponentialequation cannot be expressed with the samebase? Ex. 9x = 14
1. Write the log of each side log 9x = log 14
2. Use the power rule to simplify x log 9 = log 14
3. Solve for x
x log14
log9
4. Evaluate on calculator
x 1.1461280.9542425
1.202
5. Check 91.202 = 14
Aim: Exponential Equations using Logs
Course: Alg. 2 & Trig.
Alternate Method - 1
Solve for x to the nearest 10th:12 • 12x = 500
Method 1
log (12 • 12x) = log 500
log 12 + log 12x = log 500
log 12 + x log 12 = log 500
x log 12 = log 500 - log 12
x log500 log 12
log12
x = 1.5 to nearest 10th
Aim: Exponential Equations using Logs
Course: Alg. 2 & Trig.
Alternate Method - 2
Solve for x to the nearest 10th:12 • 12x = 500
Method 2
12x 500
12
log12x log(500
12)
x log12 log 500 log12
x log500 log 12
log12= 1.5
Aim: Exponential Equations using Logs
Course: Alg. 2 & Trig.
Alternate Method - 3
Solve for x to the nearest 10th:12 • 12x = 500
Method 3
121 + x = 500
log 121 + x = log 500
(1 + x)log 12 = log 500
1 x log500
log12
x log500
log12 1 1.5
Aim: Exponential Equations using Logs
Course: Alg. 2 & Trig.
Problems
Solve 6x = 42 to 3 decimal places
x log 42
log 6
x 1.62324929
0.77815125042.086
log 6x = log 42 Property of Equality for Log functions
x log 6 = log 42Power Property of Logarithms
Solve 3.1a – 3 = 9.42 to 3 decimal places
Solve 9a = 2a a = 4.982a = 0
Aim: Exponential Equations using Logs
Course: Alg. 2 & Trig.
Complicated Problem
Solve 82x - 5 = 5x + 1
2x log 8 - x log 5 = log 5 + 5 log 8
x log5 5log8
2log8 log5
x 0.6990 5(0.9031)
2(0.9031) 0.69904.7095
log 82x - 5 = log 5x + 1 Property of Equality for Log functions
(2x - 5)log 8 = (x + 1)log 5Power Property of Logarithms
2x log 8 - 5 log 8 = x log 5 + 1 log 5Distributive Property
x(2 log 8 - log 5) = log 5 + 5 log 8Distributive Property
Aim: Exponential Equations using Logs
Course: Alg. 2 & Trig.
Problems
Solve 4e2x = 5 to 3 decimal places
x 1
2ln
5
4
x 1
2.2231435513 0.112
ln e2x = ln 5/4 Property of Equality for Ln functions
2x = ln 5/4Inverse Property of Logs & Expos
e2x = 5/4 Divide both sides by 4
Check: 4e2(0.112) = 5