Lots O’Logs

Remember?

First a review logs

927 32

329log27 This two

equations are equivalent.

And since logs are really just exponents, we have the laws of logs.1) Multiplying arguments

baab xxx loglog)(log

2) Dividing arguments

baba

xxx logloglog

aba xb

x log)(log 3) Arguments with an exponent

Remember?

Last week we started with the problem:

Ex 4. A bank account earns interest compounded monthly at an annual rate at 4.2%. Initially the investment was $400. When does it double in value?

t

y12

12042.01400

We got the equation…

t120035.1400800

t120035.12 But then we couldn’t get common bases.

And then…

t120035.12 We took the log of both sides…

t120035.1log2log Then the “down in front” rule

0035.1log122log t Divided by 12log1.0035

t0035.1log122log

And my calculator can do this

53.16t

A shortcut to the calculator rule

So we have seen that t120035.12 can be written as

t120035.1log2log

So we do not need to take the log of both sides. We can go to log form

t122log 0035.1 And then write t120035.1log2log

Remember that the base is on the bottom!

Lots o’ Logs

Solve for x.

402 74 x

7440log2 x

742log40log

x

743219.5 x

Since I can’t get common bases, I’m stuck in exponential form. So I go to log form.

08.343219.12

xx

Now I can use the calculator rule to change the base to 10.

Lots o’ Logs

Solve for x.

10093 41 x

10033421 x

10033 81 x

1003 9 x Since I can’t get common bases, I’m stuck in exponential form. So I go to log form.9100log3 x 9

3log100log

x919.4 x81.4x

Lots o’ Logs

Solve for x.

2.04 32 x

x324log2.0log

Since I can’t get common bases, I’m stuck in exponential form. So I go to log form.

x322.0log4

05.116.1233216.1

xx

x

Lots o’ Logs

Solve for x.

617log2 x

617log1

2 x

617log12

x x617log2

x62log17log

68.0609.4

xx

Lots o’ Logs

Solve for x.1472 125 xx

When there are 2 bases we take the log of both sides.

1472 12log5log xx Now move the exponents “down in front”

12log)14(5log)72( xx

079.1)14()699.0)(72( xx

106.15079.1893.4398.1 xx

213.10319.0 x02.32x

Lots o’ Logs

35log. 4 xa 114loglog. 55 xxbSolve these equations for x

Stuck in log form so let’s write it in exponential form.

543 x

59564564

xx

x

Notice the common bases on the left hand side. Laws of logs apply: 1)14(log5 xx

1)4(log 25 xxStuck in log form

so let’s write it in exponential form. 54 2 xx

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

0542

2

x

xx

125.1

891

xx

x

Lots o’ Logs

35log2log. 22 xxd 23log5log. 33 xxc

235log3 xx

2335 xx

9)3)(5( xx

46

0)4)(6(0242

91522

2

xorx

xxxx

xx

352log2

xx

3252

xx

852

xx

)5(852)5(

xxxx

6742

4082

xxxx

Back to Wrod There’s a problem with this word!

A Sidney Crosby rookie card was purchased in 2005 for $15.oo. Its value is set to double every 2 years. When will the card be worth $90.00?

We’ve set up equations like this before.

2)2(15x

y

2)2(1590x

Isolate the power!

2)2(6x

STUCK!

26log2

x

Base is on the bottom

22log6log x

17.52

585.2

x

x

In 5.17 years, the card is worth $90.

Back to Wrod

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

We’ve set up equations like this before.

2.8

0 21

x

Ay

2.8

00 21

101

x

AA

Isolate the power!

2.8

21

101

x

STUCK? No way!

2.81.0log 5.0

x

Base is on the bottom

2.85.0log1.0log x

24.272.8

32.3

x

xIn 27.24 minutes only 1/10th the original amount will remain.

In this casey = (1/10)Ao

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Sarah bought a computer for $2000. Its value depreciates by 18% every two years. a. By what percentage does it depreciate every year?

282.2000x

y This means its value is 82% of the last value.

08.181182.2000 2

1

yy

The values create a pattern like this:

X 0 1 2y 2000 1811.08 1640

The CR is 0.906 so its losing 9.446% every year.

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Sarah bought a computer for $2000. Its value depreciates by 18% every two years. b. When is its value $99?

282.2000x

y

282.200099x

282.495.0x

2)495.0(log 82.0x

282.0log495.0log x

yearsx

x

09.72

54.3

Back to Wrod

In Jan 2011, Ben and Josh opened saving accounts. Ben invested $250 in an account that will double every 8 years. Josh invested $600 in an account paying 6% interest per year compounded daily. When will they have the same amount of money?

Ben’s Money

8)2(250t

y

Josh’s Money

t

y365

36506.01600

When they have the same amount, the y values are equal tt 365

8

36506.01600)2(250

Now solve for ‘t’.

Back to Wrod

In Jan 2011, Ben and Josh opened saving accounts. Ben invested $250 in an account that will double every 8 years. Josh invested $600 in an account paying 6% interest per year compounded daily. When will they have the same amount of money? tt 365

8

36506.01600)2(250

tt

3658 000164384.14.2)2( We’re stuck in exponential form. With two bases, we’ll take the log of both sides. tt

3658 000164384.14.2log)2log(

tt 365000164384.14.2log)2log(8

Notice the 365t is not the exponent all of the argument. So first, the laws of logs.

Back to Wrod

In Jan 2011, Ben and Josh opened saving accounts. Ben invested $250 in an account that will double every 8 years. Josh invested $600 in an account paying 6% interest per year compounded daily. When will they have the same amount of money? tt 365000164384.14.2log)2log(

8

tt 365000164384.1log4.2log)2log(8

We’re multiplying arguments so we can add their logs.

Now the 365t can come down in front

000164384.1log3654.2log)2log(8

tt

tt 0260555969.03802.0301.08

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In Jan 2011, Ben and Josh opened saving accounts. Ben invested $250 in an account that will double every 8 years. Josh invested $600 in an account paying 6% interest per year compounded daily. When will they have the same amount of money? tt 0260555969.03802.0037625.0

3802.00260555969.0037625.0 tt

3802.00115694031.0 t

86.32t In 32.86 years, they will have the same amount.

Back to Wrod

In Jan 2011, Ben and Josh opened saving accounts. Ben invested $250 in an account that will double every 8 years. Josh invested $600 in an account paying 6% interest per year compounded daily. When will they have the same amount of money?

Ben’s Money

8)2(250t

y

Josh’s Money

t

y365

36506.01600

886.32

)2(250y )86.32(365000164384.1600y

44.4309$y 62.4308$yThese are not exactly the same because we needed to round our values