Download presentation

Presentation is loading. Please wait.

1
**5-4 Exponential & Logarithmic Equations**

Strategies and Practice

2
Objectives – Use like bases to solve exponential equations. – Use logarithms to solve exponential equations. – Use the definition of a logarithm to solve logarithmic equations. – Use the one-to-one property of logarithms to solve logarithmic equations.

3
**Use like bases to solve exponential equations**

Equal bases must have equal exponents EX: Given 3x-1 = 32x + 1 then x-1 = 2x+1 x = -2 If possible, rewrite to make bases equal EX: Given 2-x = 4x+1 rewrite 4 as 22 2-x = 22x+2 then –x=2x+2 x=-2/3 Note: Isolate function if needed 3(2x)=48 2x =16

4
You try… 1. 4x = 83 (22)x = (23)3 (2)2x = (2)9 So 2x = 9, or x = 4.5 2. 5x-2 = 25x 5x-2 = (52)x 5x-2 = (5)2x So x - 2 = 2x, or x = -2 3. 6(3x+1) = 54 3x+1 = 9 3x+1 = 32 So x + 1= 2, or x = 1 4. e–x2 = e-3x - 4 So –x2 = -3x & x2 – 3x – 4 = 0 & (x-4)(x+1) = 0 & x=4 and x = -1

5
**Exponentials of Unequal Bases**

Use logarithm (inverse function) of same base on both sides of equation Solve: ex = 72 loge ex = loge x = ln 72 4.277 Solve: x-1 = 12 log7 7x-1 = log7 x - 1 = log7 12 x = log log 12 log 7 2.277 x =

6
**You try… 1. Solve 3(2x) = 42 x = log2 14 3.807 2. Solve 32t-5 = 15**

t = 1/2(log ) 3.732 3. Solve e2x = 5 x = 1/2 ln 5 0.805 4. Solve ex + 5 = 60 x = ln 55 4.007

7
**Solving Logarithmic Equations**

Rewrite into exponential form EX: Solve: ln x = - 1/2 loge x = - 1/2 e -1/2 = x 0.607 x = e -1/2 EX: Solve: log5 3x = 4 log5 3x = 2 52= 3x 25= 3x 25/3= x x = 25/3 8.333

8
**Solving Logarithmic Equations**

Use properties of logarithms to condense. EX: Solve: log4x + log4(x-1) = ½ log4 x(x – 1) = 1/2 Check for extraneous roots. 4 1/2 = x(x – 1) 2 = x2 – x 0 = x2 – x – 2 0 = (x – 2)(x + 1) x = 2 & x = -1

9
**You try… 1. Solve ln x = -7 x = e-7 0.000912 2. Solve 2 log3 2x = 4**

3. Solve ln x + ln (x-3) = 0 3 + 13 2 3 - 13 2 x = & x = 4. Solve ln x = 4 x = e-1/2 0.607

10
**Double-Sided Log Equations**

Equate powers (domain solutions only) EX: Solve: log5(5x-1) = log5(x+7) 5x – 1 = x + 7 4x = 8 x = 2 EX: Solve: ln(x-2) + ln(2x-3) = 2lnx ln (x-2)(2x-3) = ln x2 Use the properties to condense. Check for extraneous roots. (x-2)(2x-3) = x2 2x2 – 7x + 6 = x2 x2 – 7x + 6 = 0 (x – 6)(x – 1)= 0 x = 6 & x = 1

11
**You try… 1. Solve ln3x2 = lnx x = 0 & x = 1/3**

2. Solve log6(3x + 14) – log6 5 = log6 2x x = 2 3. Solve log2x+log2(x+5) = log2(x+4) x = 2 & x = 2

12
**SUMMARY Equal bases Equal exponents**

Unequal bases Apply log of given base Single side logs Convert to exp form Double-sided logs Equate powers Note: Any solutions that result in a log(neg) cannot be used!

Similar presentations

© 2024 SlidePlayer.com Inc.

All rights reserved.

To make this website work, we log user data and share it with processors. To use this website, you must agree to our Privacy Policy, including cookie policy.

Ads by Google