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 b>0 & b≠1 if bx = by, then x=y

Exponential Equations

Solve by equating exponents

• 43x = 8x+1

• (22)3x = (23)x+1 rewrite w/ same base

• 26x = 23x+3

• 6x = 3x+3

• x = 1

Check → 43*1 = 81+1

64 = 64

Your turn!

• 24x = 32x-1

• 24x = (25)x-1

• 4x = 5x-5

• 5 = xBe sure to check your answer!!!

When you can’t rewrite using the same base, you can solve by taking

a log of both sides

• 2x = 7

• log 2x = log 7

• x log 2 = log 7

• x = ≈ 2.8072log

7log

4x = 15

• log 4x = log 15

• x log 4 = log15

• x = log15/log4

• ≈ 1.953

102x-3+4 = 21• -4 -4• 102x-3 = 17• log 102x-3 = log 17• (2x-3) log 10 = log 17• 2x – 3 = log17/log 10• 2x = (log 17/log10) +3• ≈ 2.115

5x+2 + 3 = 25• 5x+2 = 22• log 5x+2 = log 22• (x+2) log 5 = log 22• x+2 = (log 22/log 5)• x = (log22/log5) – 2• ≈ -.079

Solving Log Equations

• To solve use the property for logs w/ the same base:

• + #’s b,x,y & b≠1

• If logbx = logby, then x = y

log3(5x-1) = log3(x+7)

•5x – 1 = x + 7• 5x = x + 8• 4x = 8• x = 2 and check• log3(5*2-1) = log3(2+7)• log39 = log39

When you have a log on 1 side only, rewrite in exponential form

• b>0 & b≠1

• if logbx = y, then by = x

log5(3x + 1) = 2

• (3x+1) = 52

• 3x+1 = 25

• x = 8 and check

• Because the domain of log functions doesn’t include all reals, you should check for extraneous solutions

log23 + log2(x-7) = 3

• log2 3(x-7) = 3• log2 (3x - 21) = 3• 23 = 3x – 21 • 8 = 3x – 21• 29 = 3x• 29/3 = x

log5x + log(x+1)=2• log (5x)(x+1) = 2 (product property)

• log (5x2 – 5x) = 2

• 5x2 - 5x = 102 (rewrite)

• 5x2 - 5x = 100

• x2 – 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

Homework:

• Page 505– # 26, 28, 33, 35, 45, 50, 53, 56

Answers to HW p. 505

26) x = -1/5 28) no sol

33) x = 3/2 35) x = 0.350

45) x = 2187 50) x = 15

53) No sol. (x = 2 but it doesn’t work when you plug it in and check)

56) No sol. (x = 2 but it doesn’t work when you plug it in and check)

Natural Exponential Function

y = ex

Natural Base

ln e = 1

Natural Logarithmic Function

y = ex x = loge y

x = ln y

To solve natural log functions, you solve them the same as logarithmic

functions.

Remember:ln has base e

ln e = 1

1.) ex+7 = 98

2.) 4e3x-5 = 72

3.) ln x3 - 5 = 1

5 – ln x = 7

5 – ln x = 7

- ln x = 2

ln x = -2

e2 = x

7.389 = x

3e-x – 4 = 9

3e-x = 13

e-x = 13/3

ln e-x = ln (13/3)

-x ln e = 1.466

x = -1.466

Homework

• Page 505– #32, 34, 40, 43, 46, 48