9.5 Exponential Equations & Inequalities

Logarithmic vocabulary

Consider:

log 260

Also:

log 0.26

Ex 1) Underline the mantissa & circle the characteristic

log 425 = 2.6284

If we are given log x or ln x, we can find x using our calculators. We will use 10x or ex.

Let’s practice some simple ones…get those calculators ready!

= log (2.6 × 102) = log 2.6 + log 102 = 0.4150 + 2 = 2.4150

= log (2.6 × 10–1) = log 2.6 + log 10–1= 0.4150 + –1 = –0.5850

mantissa characteristic

Ex 2) Solve for x to the nearest hundredth. a) log x = 3.2274

103.2274 = x

1688.11 = x

b) 2 log x = 2.6419

log x = 1.32095

101.32095 = x

20.94 = x

c) ln x – 3 = 5.7213

ln x = 8.7213

e8.7213 = x

6132.15 = x

To solve an exponential equation:

(1) Isolate the exponential expression

(2) Take the logarithm of both sides of the equation

(3) Verify all answers! (by substitution in original)

Ex 3) Solve for x to nearest hundredth.

42x – 1 – 27 = 0

42x – 1 = 27

log 42x – 1 = log 27

(2x – 1) log 4 = log 27

2x log 4 – log 4 = log 27

2x log 4 = log 27 + log 4

log 27 + log 4 2 log 4

x = 1.69

Ex 4)

54

54

54

51

5 5

5 4

ln ln

ln

0.22

x

x

x x

x

x

x

e

e

e e

e

e

e

x

x

x =

Sometimes we can’t solve algebraically, so we go to our graphing calculator.

Solve using a graphing calculator.

Ex 5) ex = x2 – 1

Y1 = ex

Y2 = x2 – 1

(Find intersection)

x = –1.15

Ex 6) y ≥ ex – 2

Y1 = ex – 2

Applications

Compound Interest Formula:

A = total value of investment

t = number of years

P = principal amount invested

r = interest rate

n = number of times per year interest is compounded

1nt

rA P

n

(% decimal)

Ex 7) The Smith Family wants to give their youngest daughter $20,000 when she is ready for college. They now have $11,500 to invest. Determine how many years it will take them to achieve their goal given that they invest this amount at 8.3% compounded monthly.

12

12

.08312

log log

1

.08320,000 11,500 1

12

20,000 .0831

11,500 12

.083log 20,000 log11,500 12 log 1

12log 20,000 log11,500

12log 1

ntrn

t

t

A P

t

t

A = 20,000P = 11,500r = .083n = 12

*Watch those parentheses!

t = 7 years

Continuous Compound Interest Formula A = Pert

Ex 8) A sum of money invested at a fixed interest rate, compounded continuously, tripled in 19 years. Determine the interest rate at which the money was invested.

A = Pert

3P = Per(19)

P P 3 = e 19r

ln 3 = 19r lneln 319 r = 5.8%

*you don’t know A or P but you don’t need it!You need P to triple

= r

Homework

#906 Pg 472 #1, 3, 5, 9, 13, 18, 20–23, 25, 27, 29, 32, 33, 36, 38, 39–47 odd