Logs – Part 2

Review of Logarithms

is... form clogarithmi in ...writenlog

log is... form clogarithmi in ...writen

cbac

accb

ab

ba

baab xxx loglog)(log

baba

xxx logloglog

aba xb

x log)(log

baxabx

loglog then if

abba log

loglog

abab log

3 logarithm laws3 logarithm shortcuts

Solving log equationsSolving logarithmic equations takes some instinct, which only comes from practice, but to help you get you started, here is a flowchart with some possibly useful steps.

Simplify:evaluate any

complete log or exponential expressions

Isolate the unknown:

If the unknown is in

the…

…argument: change it to exponential form

…exponent: in exponential form get common bases if possible, or change to logarithmic form to solve, or take the log of both sides and apply log

rules

…base: write in exponential form then remove the exponent by raising each

side to the opposite exponent

Solving Exponential/Logarithmic Equations Example

Ex: Solve for x.1472 125 xx

1472 12log5log xx

12log)14(5log)72( xx079.1)14()699.0)(72( xx

106.15079.1893.4398.1 xx213.10319.0 x02.32x

35loga. 4 x 114loglogb. 55 xx

Solve these equations for x

543 x

59564

564

xxx

114log5 xx

1)4(log 25 xx

12 54 xx

054 2 xx

1or25.1 xx

54 2 xx

891

)4(2)5)(4(4)1()1( 2

x

x

Solving Exponential/Logarithmic Equations Practice

Lots o’ Logs 35log2logd. 22 xx 23log5logc. 33 xx

235log3 xx

2335 xx

9)3)(5( xx

0)4)(6(0242

91522

2

xxxxxx

352log2

xx

3252

xx

852

xx

)5(852)5(

xxxx

6742

4082

xxxx

46 xorx

Applications - LogarithmsEx 1. A Sidney Crosby rookie card was purchased in 2005 for $15.00. Its value is set to double every 2 years. When will the card be worth $90.00?

2)2(15x

y

2)2(1590x

2)2(6x

22log6log x

17.52

585.2

x

x

In 5.17 years, the card is worth $90.

Ex 2. A certain radioactive element has a half-life of 8.2 minutes. When will there be 1/10th the original amount?

2.8

0 21

x

Ay

2.8

00 21

101

x

AA

2.8

21

101

x

2.85.0log1.0log x

24.272.8

32.3

x

x

It will take 27.24 minutes for only 1/10th the original amount to remain.

In this casey = (1/10)Ao

2.85.01.0x

Applications - Logarithms

Ex 3. Sarah bought a computer for $2000. Its value depreciates by 18% every two years.

282.2000x

y r = 1 – 0.18 = 0.82This means it will be worth 82% of its value after 2 years.

08.181182.2000 2

1

yy

In one year, it went from being worth $2000 to being worth $1811.08. Dividing tells us that it is 90.554% of $2000, or a depreciation of 9.446% in one year.

a. By what percentage does it depreciate every year?

Applications - Logarithms

b. When is its value $99?

282.2000x

y

282.200099x

282.0495.0x

282.0log0495.0log x

29.302

146.15

x

xIn 30.29 years her computer will be worth $99.

Applications - LogarithmsEx 3. Sarah bought a computer for $2000. Its value depreciates by 18% every two years.