Remember that exponential functions and logarithmic functions are inverses of each other. We will use this property to solve problems.

Slides:

Advertisements

Similar presentations

We now know that a logarithm is perhaps best understood as being closely related to an exponential equation. In fact, whenever we get stuck in the problems.

Advertisements

Essential Question: What are some of the similarities and differences between natural and common logarithms.

MAT 150 – Class #19. Objectives  Solve an exponential equation by writing it in logarithmic form  Convert logarithms using the change of base formula.

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.

Properties of Logarithms

Solve an equation with variables on both sides

7.5 E XPONENTIAL AND L OGARITHMIC E QUATIONS. E XPONENTIAL E QUATIONS An exponential equation is an equation containing one or more expressions that have.

Solve an absolute value equation EXAMPLE 2 SOLUTION Rewrite the absolute value equation as two equations. Then solve each equation separately. x – 3 =

Standardized Test Practice

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.

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

4.6 Solve Exponential and Logarithmic Equations

5-4 Exponential & Logarithmic Equations

7.6 – Solve Exponential and Log Equations

7.5 E XPONENTIAL AND L OGARITHMIC E QUATIONS AND I NEQUALITIES Solve exponential and logarithmic equations and equalities. Solve problems involving exponential.

Objectives Solve exponential and logarithmic equations and equalities.

Warm-up Find the domain and range from these 3 equations.

8 – 6 Solving Exponential and Logarithmic Equations Day2 Objective: CA. 11.1: Students understand the inverse relationship between exponents and logarithms.

EQ: How do you use the properties of exponents and logarithms to solve equations?

Algebra II w/trig. A logarithm is another way to write an exponential. A log is the inverse of an exponential. Definition of Log function: The logarithmic.

Academy Algebra II/Trig 6.6: Solve Exponential and Logarithmic Equations Unit 8 Test ( ): Friday 3/22.

1. Expand the following: 2. Condense the following: Warm-upWarm-up.

( ) EXAMPLE 3 Standardized Test Practice SOLUTION 5 x = – 9 – 9

Notes Over 8.4 Rewriting Logarithmic Equations Rewrite the equation in exponential form.

Algebra – S OLVING R ATIONAL E QUATIONS. O BJECTIVES  Revisit concepts of least common denominators  Solve rational equations  Understand and.

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.

Solving Systems of Equations: The Elimination Method Solving Systems of Equations: The Elimination Method Solving Systems of Equations: The Elimination.

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.

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 =

Use the substitution method

Solving Logarithmic Equations

S OLVING E XPONENTIAL AND LOGARITHMIC E QUATIONS Unit 3C Day 5.

Exponential and Logarithmic Equations

Solving Equations Using Addition or Subtraction Objective: Students will solve linear equations using addition and subtraction.

Lesson 4 Contents 11-3 Solving Quadratic Equations by Using the Quadratic Formula Objectives 1. Solve quadratic equations by using the Quadratic Formula.

4.7 (Green) Solve Exponential and Logarithmic Equations No School: Monday Logarithms Test: 1/21/10 (Thursday)

Solving Logarithmic Functions Math 3 Standard MM3A2.

Solving Exponential and Logarithmic Equations Section 3.4.

Holt McDougal Algebra Exponential and Logarithmic Equations and Inequalities 4-5 Exponential and Logarithmic Equations and Inequalities Holt Algebra.

10.3 Solving Quadratic Equations – Solving Quadratic Eq. Goals / “I can…”  Solve quadratic equations by graphing  Solve quadratic equations using.

Solving Logarithmic Equations I.. Relationship between Exponential and Logarithmic Equations. A) Logs and Exponentials are INVERSES of each other. 1) That.

3.4 Solving Exponential and Logarithmic Equations.

S OLVING E XPONENTIAL AND LOGARITHMIC E QUATIONS Unit 3C Day 5.

Example 1 Solve Using Equal Powers Property Solve the equation. a. 4 9x = – 4 x x23x = b. Write original equation. SOLUTION a. 4 9x 5 42.

For b > 0 and b  1, if b x = b y, then x = y.

Rewrite a linear equation

Ch. 8.5 Exponential and Logarithmic Equations

1. Add: 5 x2 – 1 + 2x x2 + 5x – 6 ANSWERS 2x2 +7x + 30

Solving Exponential Equations

Solving Logarithmic Equations

3.4 Quick Review Express In 56 in terms of ln 2 and ln 7.

Solving Exponential and Logarithmic Functions

( ) EXAMPLE 3 Standardized Test Practice SOLUTION 5 x = – 9 – 9

Solving Logarithmic Equations

Solving Exponential Equations

8.6 Solving Exponential & 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.

7.6 Solve Exponential and Logarithmic Equations

Solving one- and two-step equations

Objectives Use properties to simplify logarithmic expressions.

Keeper #39 Solving Logarithmic Equations and Inequalities

Solve an inequality using subtraction

EXPONENTIAL FUNCTION where (base) b > 0 and b For 0 < b < 1,

For b > 0 and b ≠ 1, if b x = b y, then x = y.

3-2 Solving Inequalities Using Addition and Subtraction

Example 2B: Solving Linear Systems by Elimination

Definition of logarithm

Presentation transcript:

Remember that exponential functions and logarithmic functions are inverses of each other. We will use this property to solve problems.

S OLVING E XPONENTIAL E QUATIONS x = 5

Solving by Equating Exponents Solve 4 3x = 8 x + 1. ( 2 2 ) 3x = ( 2 3 ) x + 1 6x = 3x x = 2 3x (3x) = 2 3(x + 1) x = 1 The solution is 1. S OLUTION Rewrite each power with base 2. Power of a power property Equate exponents. Solve for x. 4 3 x = 8 x + 1 Write original equation. C HECK Check the solution by substituting it into the original equation = = 64 Solution checks. Solve for x.

Solving by Equating Exponents When it is not convenient to write each side of an exponential equation using the same base, you can solve the equation by taking a logarithm of each side.

Taking a Logarithm of Each Side Solve 10 2 x – = x – 3 = 17 log 10 2 x – 3 = log 17 2 x = x = ( ) 1 2 x  Use a calculator x – = 21 S OLUTION Write original equation. Subtract 4 from each side. Add 3 to each side. Multiply each side by. 1 2 Take log base 10 of each side. 2 x – 3 = log 17 log 10 x = x

C HECK Taking a Logarithm of Each Side Check the solution algebraically by substituting into the original equation. Solve 10 2 x – = 21.

S OLVING L OGARITHMIC E QUATIONS To solve a logarithmic equation, use this property for logarithms with the same base: For positive numbers b, x, and y where b  1, log b x = log b y if and only if x = y.

Use property for logarithms with the same base. 5 x = x + 8 Solving a Logarithmic Equation Solve log 3 (5 x – 1) = log 3 (x + 7). 5 x – 1 = x + 7 x = 2 The solution is 2. S OLUTION Use inverse property with base 3. Add 1 to each side. Solve for x.

Use property for logarithms with the same base. 5 x = x + 8 Solving a Logarithmic Equation 5 x – 1 = x + 7 x = 2 S OLUTION log 3 (5 x – 1) = log 3 (x + 7) Write original equation. Add 1 to each side. Solve for x. C HECK Check the solution by substituting it into the original equation. log 3 (5 x – 1) = log 3 (x + 7) log 3 9 = log 3 9 Solution checks. log 3 (5 · 2 – 1) = log 3 (2 + 7) ? Write original equation. Substitute 2 for x.

log 5 (3x + 1) = 2 Solving a Logarithmic Equation Solve log 5 (3x + 1) = 2. 3x + 1 = 25 x = 8 The solution is 8. Write original equation. Exponentiate each side Solve for x. Simplify.

log 5 (3x + 1) = 2 Solving a Logarithmic Equation Solve log 5 (3x + 1) = 2. 5 = 5 2 log 5 (3x + 1) 3x + 1 = 25 x = 8 The solution is 8. S OLUTION Write original equation. Exponentiate each side using base 5. b = x log b x Solve for x. log 5 (3x + 1) = 2 log 5 (3 · 8 + 1) = 2 ? log 5 25 = 2 ? 2 = 2 Solution checks. C HECK Check the solution by substituting it into the original equation. Simplify. Substitute 8 for x. Write original equation.

Because the domain of a logarithmic function generally does not include all real numbers, you should be sure to check for extraneous solutions of logarithmic equations. You can do this algebraically or graphically. Checking for Extraneous Solutions

Solve log 5 x + log (x – 1) = 2. Check for extraneous solutions. log [ 5 x (x – 1)] = 2 5 x 2 – 5 x = 100 x 2 – x – 20 = 0 (x – 5 )(x + 4) = 0 x = 5 or x = – 4 S OLUTION log 5 x + log (x – 1) = 2 Write original equation. Product property of logarithms. Exponentiate both sides. Write in standard form. Factor. Zero product property Checking for Extraneous Solutions Simplify

The solutions appear to be 5 and – 4. However, when you check these in the original equation you can see that x = 5 is the only solution. S OLUTION log 5 x + log (x – 1) = 2 Check for extraneous solutions. x = 5 or x = – 4 Zero product property Checking for Extraneous Solutions Check: log 5(5) + log(5 – 1) = 2 log 5(-4) + log(-4 – 1) = 2 log 25 + log 4 = 2 log log -5 = 2 log (25)(4) = 2 error log 100 = 2 2 = 2

If necessary use the properties of logarithms to condense several terms into one. If necessary use the properties of logarithms to condense several terms into one. S OLVING L OGARITHMIC E QUATIONS Make sure that there is one term on each side of equation. Always use the inverse property.