7.6A Solving Exponential and Logarithmic Equations Algebra II.

Slides:



Advertisements
Similar presentations
Solve Exponential Equations
Advertisements

Solving Exponential and Logarithmic Equations
Solving equations involving exponents and logarithms
Warm-Up. One way to solve exponential equations is to use the property that if 2 powers w/ the same base are equal, then their exponents are equal. For.
8.6 Solving Exponential and Logarithmic Equations p. 501.
Evaluating logarithms
CH. 8.6 Natural Logarithms. Write 2 ln 12 – ln 9 as a single natural logarithm. 2 ln 12 – ln 9 = ln 12 2 – ln 9Power Property = lnQuotient Property 12.
Questions over 4.6 HW???. 4.7 (Green) Solve Exponential and Logarithmic Equations No School: Monday Logarithms Test: 1/21/10 (Thursday)
5.4 Exponential and Logarithmic Equations Essential Questions: How do we solve exponential and logarithmic equations?
Exponential and Logarithmic Equations
7-5 Logarithmic & Exponential Equations
Take a logarithm of each side
4.6 Solve Exponential and Logarithmic Equations
5-4 Exponential & Logarithmic Equations
7.6 – Solve Exponential and Log Equations
Solving Exponential and Logarithmic Equations
5.6 Laws of Logarithms 5.7 Exponential Equations; Changing Base
Objectives Solve exponential and logarithmic equations and equalities.
Logarithmic and Exponential Equations
8.6 Solving Exponential and Logarithmic Equations
5.7 – Exponential Equations. 5.7 Exponential Equations Objectives:  Solve Exponential Equations using the Change of Base Formula  Evaluate logarithms.
Quiz 7-5: Expand Condense Use these to find: Use change of base to solve 7.
8 – 6 Solving Exponential and Logarithmic Equations Day 1 Objective: Solve exponential equations.
Solving Exponential and Logarithmic Equations Section 8.6.
MAT 171 Precalculus Algebra T rigsted - Pilot Test Dr. Claude Moore - Cape Fear Community College CHAPTER 5: Exponential and Logarithmic Functions and.
Solving Logarithmic Equations
Solve a logarithmic equation
EXAMPLE 4 Solve a logarithmic equation Solve log (4x – 7) = log (x + 5). 5 5 log (4x – 7) = log (x + 5) x – 7 = x x – 7 = 5 3x = 12 x = 4 Write.
Do Now (7.4 Practice): Graph. Determine domain and range.
Aim: Exponential Equations using Logs Course: Alg. 2 & Trig. Aim: How do we solve exponential equations using logarithms? Do Now:
Logarithms 1 Converting from Logarithmic Form to Exponential Form and Back 2 Solving Logarithmic Equations & Inequalities 3 Practice Problems.
Solving Exponential Equations. We can solve exponential equations using logarithms. By converting to a logarithm, we can move the variable from the exponent.
For b > 0 and b ≠ 1, if b x = b y, then x = y. S OLVING E XPONENTIAL E QUATIONS If two powers with the same base are equal, then their exponents must be.
CP Math 10-5 Solve Exponential and Logarithmic Equations.
EXAMPLE 1 Solve by equating exponents Rewrite 4 and as powers with base Solve 4 = x 1 2 x – 3 (2 ) = (2 ) 2 x – 3x – 1– 1 2 = 2 2 x– x + 3 2x =
Solving Logarithmic Equations
EXAMPLE 2 Take a logarithm of each side Solve 4 = 11. x 4 = 11 x log 4 = log 11 x 4 4 log 4 x = 11 x = log 11 log 4 x 1.73 SOLUTION Write original equation.
4.7 (Green) Solve Exponential and Logarithmic Equations No School: Monday Logarithms Test: 1/21/10 (Thursday)
Solving Exponential and Logarithmic Equations Section 3.4.
Topic 10 : Exponential and Logarithmic Functions Solving Exponential and Logarithmic Equations.
Topic 10 : Exponential and Logarithmic Functions Solving Exponential and Logarithmic Equations.
3.4 Solving Exponential and Logarithmic Equations.
For b > 0 and b  1, if b x = b y, then x = y.
Ch. 8.5 Exponential and Logarithmic Equations
Section 3.4 Solving Exponential and Logarithmic Equations
Solving Exponential and Logarithmic Equations Day 7 AB Calculus Precalculus Review Hopkins.
Solving Exponential and Logarithmic Equations
Exponential Equations
Exponential and Logarithmic Equations
Modeling.
Exponential & Logarithmic Equations
Solving Exponential and Logarithmic Equations
Today in Precalculus Go over homework
7.5 Exponential and Logarithmic Equations
8.6 Solving Exponential and Logarithmic Equations
8.6 Solving Exponential and Logarithmic Equations
Essential Question: How do I graph & solve exponential and logarithmic functions? Daily Question: What are the properties of logarithms and how do I use.
Day “35” Solving Exponential and Logarithmic Equations
7.6 Solve Exponential and Logarithmic Equations
Worksheet Key 1/2/2019 9:28 PM Solving Exp and Log Equations.
WARMUP Lesson 7.6, For use with pages
Section 5.5 Additional Popper 34: Choice A for #1 – 10
Logarithmic and Exponential Equations
8.6 Solving Exponential and Logarithmic Equations
Logarithmic and Exponential Equations
Section 6.6 Solving Exponential and Logarithmic Equations
For b > 0 and b ≠ 1, if b x = b y, then x = y.
Unit 3: Exponential and Logarithmic Functions
Section 5.5 Additional Popper 34: Choice A for #1 – 10
Warm Ups.
Presentation transcript:

7.6A Solving Exponential and Logarithmic Equations Algebra II

One way to solve exponential equations is to use the property that if 2 powers w/ the same base are equal, then their exponents are equal. For b>0 & b≠1 if b x = b y, then x=y Exponential Equations

Ex. 1)Solve by equating exponents 4 3x = 8 x+1 (2 2 ) 3x = (2 3 ) x+1 rewrite w/ same base 2 6x = 2 3x+3 6x = 3x+3 x = 1 Check → 4 3*1 = = 64

Ex. 2) Solve by equating exponents 2 4x = 32 x-1 2 4x = (2 5 ) x-1 4x = 5x-5 5 = x Be sure to check your answer!!!

Ex. 3 When you can’t rewrite using the same base, you can solve by taking a log of both sides 2 x = 7 log 2 2 x = log 2 7 x = log 2 7 x = ≈ 2.807

Ex. 4) Solve by equating exponents log 4 4 x = log 4 15 x = log 4 15 = log15/log4 ≈ 1.953

Ex. 5 5 x = 25 5 x+2 = 22 log 5 5 x+2 = log 5 22 x+2 = log 5 22 x = (log 5 22) – 2 = (log22/log5) – 2 ≈ -.079

Ex x-3 +4 = x-3 = 17 log x-3 = log x-3 = log 17 2x = 3 + log17 x = ½(3 + log17) ≈ 2.115

Assignment

7.6B Solving Log Equations To solve use the property for logs w/ the same base: + #’s b,x,y & b≠1 If log b x = log b y, then x = y

Ex. 1 log 3 (5x-1) = log 3 (x+7) 5x – 1 = x + 7 5x = x + 8 4x = 8 x = 2 and check log 3 (5*2-1) = log 3 (2+7) log 3 9 = log 3 9

When faced with log/logs on one side of equation, then exponentiate each side. b>0 & b≠1 if x = y, then b x = b y

Ex. 2) log 5 (3x + 1) = 2 5 log 5 (3x+1) = 5 2 3x+1 = 25 x = 8 and check Because the domain of log functions doesn’t include all reals, you should check for extraneous solutions

Ex. 3 log 2 x + log 2 (x-7) = 3 log 2 x(x-7) = 3 log 2 (x 2 - 7x) = 3 2 log 2 x² -7x = x 2 – 7x = 8 x 2 – 7x – 8 = 0 (x-8)(x+1)=0 x=8 x= -1

Ex. 4 log5x + log(x-1)=2 log (5x)(x-1) = 2 (product property) log (5x 2 – 5x) = 2 10 log5x -5x = x 2 - 5x = 100 x 2 – x - 20 = 0 (subtract 100 and divide by 5) (x-5)(x+4) = 0 x=5, x=-4 graph and you’ll see 5=x is the only solution 2

EX. 5 Newton’s Law of Cooling The temperature T of a cooling time t (in minutes) is: T = (T 0 – T R ) e -rt + T R T 0 = initial temperature T R = room temperature r = constant cooling rate of the substance

You’re cooking stew. When you take it off the stove the temp. is 212°F. The room temp. is 70°F and the cooling rate of the stew is r =.046. How long will it take to cool the stew to a serving temp. of 100°?

T 0 = 212, T R = 70, T = 100 r =.046 So solve: 100 = (212 – 70)e -.046t = 142e -.046t (subtract 70).221 ≈ e -.046t (divide by 142) How do you get the variable out of the exponent?

ln.221 ≈ ln e -.046t (take the ln of both sides) ln.221 ≈ -.046t ≈ -.046t 33.8 ≈ t about 34 minutes to cool! Cooling cont.