7-5 Logarithmic & Exponential Equations

Slides:

Advertisements

Similar presentations

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.

Advertisements

8.6 Solving Exponential and Logarithmic Equations p. 501.

EXAMPLE 4 Solve proportions SOLUTION a x 16 = Multiply. Divide each side by 10. a x 16 = = 10 x5 16 = 10 x80 = x8 Write original proportion.

Solve an equation with variables on both sides

EXAMPLE 1 Solve a simple absolute value equation Solve |x – 5| = 7. Graph the solution. SOLUTION | x – 5 | = 7 x – 5 = – 7 or x – 5 = 7 x = 5 – 7 or x.

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

Solve an equation using subtraction EXAMPLE 1 Solve x + 7 = 4. x + 7 = 4x + 7 = 4 Write original equation. x + 7 – 7 = 4 – 7 Use subtraction property of.

EXAMPLE 3 Solve an equation by factoring Solve 2x 2 + 8x = 0. 2x 2 + 8x = 0 2x(x + 4) = 0 2x = 0 x = 0 or x + 4 = 0 or x = – 4 ANSWER The solutions of.

Standardized Test Practice

Solve a radical equation

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

and Logarithmic 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

Objectives Solve exponential and logarithmic equations and equalities.

Logarithmic and Exponential Equations

Table of Contents Solving Exponential Equations An exponential equation is an equation with a variable as part of an exponent. The following examples will.

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

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

11.3 – Exponential and Logarithmic Equations. CHANGE OF BASE FORMULA Ex: Rewrite log 5 15 using the change of base formula.

8.5 – Exponential and Logarithmic Equations. CHANGE OF BASE FORMULA where M, b, and c are positive numbers and b, c do not equal one. Ex: Rewrite log.

Solving Exponential and Logarithmic Equations Section 8.6.

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:

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

Logarithms 1 Converting from Logarithmic Form to Exponential Form and Back 2 Solving Logarithmic Equations & Inequalities 3 Practice Problems.

Chapter solving exponential and logarithmic functions.

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.

Solve an equation using addition EXAMPLE 2 Solve x – 12 = 3. Horizontal format Vertical format x– 12 = 3 Write original equation. x – 12 = 3 Add 12 to.

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 =

Example 1 Solving Two-Step Equations SOLUTION a. 12x2x + 5 = Write original equation. 112x2x + – = 15 – Subtract 1 from each side. (Subtraction property.

Use the substitution method

Solving Logarithmic Equations

Start Up Day What is the logarithmic form of 144 = 122?

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)

Property of Logarithms If x > 0, y > 0, a > 0, and a ≠ 1, then x = y if and only if log a x = log a y.

Solving Exponential Equations using Logs Wednesday, February 10, 2016 Topic 7-9 in TEXT.

7.6A Solving Exponential and Logarithmic Equations Algebra II.

Topic 10 : Exponential and Logarithmic Functions Solving Exponential and Logarithmic Equations.

3.4 Solving Exponential and Logarithmic Equations.

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.

Review of Logarithms. Review of Inverse Functions Find the inverse function of f(x) = 3x – 4. Find the inverse function of f(x) = (x – 3) Steps.

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

8.5 – Exponential and Logarithmic Equations

Ch. 8.5 Exponential and Logarithmic Equations

Solving Exponential and Logarithmic Equations

Exponential Equations

8.5 – Exponential and Logarithmic Equations

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

8.6 Solving Exponential & Logarithmic Equations

Change of Base.

Exponential & Logarithmic Equations

7.5 Exponential and Logarithmic Equations

7.6 Solve Exponential and Logarithmic Equations

Logarithmic and Exponential Equations

Logarithmic and Exponential Equations

3.4 Exponential and Logarithmic Equations

Solve an inequality using subtraction

Solving Logarithmic Equations

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

Compound Interest If a principal P is invested at an interest rate r for a period of t years, then the amount A of the investment is given by A = P(1 +

Definition of logarithm

Presentation transcript:

7-5 Logarithmic & Exponential Equations

Terms and Concepts these will be on a quiz

Write original equation. EXAMPLE 1 Solve by equating exponents Solve 4 = x 1 2 x – 3 SOLUTION 4 = x 1 2 x – 3 Write original equation. Rewrite 4 and as powers with base 2. 1 2 (2 ) = (2 ) 2 x – 3 x – 1 2 = 2 2x – x + 3 Power of a power property 2x = –x + 3 Property of equality for exponential equations x = 1 Solve for x. The solution is 1. ANSWER

GUIDED PRACTICE for Example 1 Solve the equation. 3. 81 = 1 3 3. 81 = 3 – x 1 3 5x – 6 1. 9 = 27 2x x – 1 SOLUTION –3 SOLUTION –6 2. 100 = 1000 7x + 1 3x – 2 – 8 5 SOLUTION

Write original equation. EXAMPLE 2 Take a logarithm of each side Solve 4 = 11. x SOLUTION 4 = 11 x Write original equation. log 4x = log 4 11 4 Take log of each side. 4 log 4 x = 11 log b = x b x x = log 11 log 4 Change-of-base formula x 1.73 Use a calculator. The solution is about 1.73. Check this in the original equation. ANSWER

GUIDED PRACTICE for Examples 2 and 3 Solve the equation. 4. 2 = 5 4. 2 = 5 x SOLUTION about 2.32 5. 7 = 15 9x SOLUTION about 0.155

Terms and Concepts these will be on a quiz

Solve a logarithmic equation EXAMPLE 4 Solve a logarithmic equation Solve log (4x – 7) = log (x + 5). 5 SOLUTION log (4x – 7) = log (x + 5). 5 Write original equation. 4x – 7 = x + 5 Property of equality for logarithmic equations 3x – 7 = 5 Subtract x from each side. 3x = 12 Add 7 to each side. x = 4 Divide each side by 3. The solution is 4. ANSWER

Exponentiate each side of an equation EXAMPLE 5 Exponentiate each side of an equation Solve (5x – 1)= 3 log 4 SOLUTION (5x – 1)= (5x – 1)= 3 log 4 Write original equation. 4log4(5x – 1) = 4 3 Exponentiate each side using base 4. b = x log b x 5x – 1 = 64 5x = 65 Add 1 to each side. x = 13 Divide each side by 5. The solution is 13. ANSWER

Solve the equation. Check for extraneous solutions. GUIDED PRACTICE for Examples 4, 5 and 6 Solve the equation. Check for extraneous solutions. 7. ln (7x – 4) = ln (2x + 11) 9. log 5x + log (x – 1) = 2 SOLUTION 5 SOLUTION 3 8. log (x – 6) = 5 2 10. log (x + 12) + log x =3 4 SOLUTION 4 SOLUTION 38