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What is a Log Equation:~ A log equation is a different way of

writing an exponential equation.For example: 420=5 can be written as

log45=20

The Way this works is: •You write log•You put the base # is subscript•Then write the # that the original equation is equal to•Put in the = sign•Write the exponent

log4 5=20

Adding logsWhen two or more logs with like bases are added, you multiply the numbers inside the log.log (5)+ log (7) ln(x+3)+ln(x-2)

log 35

=1.544

ln(x +x-6)2

Subtracting logsWhen two or more logs with like bases are

subtracted, you divide the numbers inside the log.

log(x+7)-log(x)

log( )x+7x

ln(360)-ln(15)

36015

ln(24)

ln( )

=3.178

Numbers in front of logs are the same thing as having the number inside the log to a power. so...

1\2ln(25) ln( 25) ln 5 1.609√

Numbers in front of logs

2log (x+1) is the same as log (x+1)5 5

Examples2*5x/4 = 250

2

log5125=x/4

4(log5125)=x

x=12

-Divide both sides by 2-Write as a logarithmic equation.-Multiply Both Sides by 4

-Simplify

Exampleslog(3x)= 15

7

10 =3x157

3

x= 46.317

-Re-write as an exponential equation-Divide both sides by 3

Examples

ln(x-3)-ln(x)=ln(7)

ln =ln7(x-3)x

x-3=7x

6x=-36

x= -12

-Subtract Log's

-Cancel out ln's

-Subtract x from both sides-Divide both sides by 6

-Simplify

log (x+5)+log (x+2)=log (x+6)2 2 2

(x+5)(x+2)=(x+6) -In this situation, because all of the log bases were the same, they cancel, leaving just the numbers inside the logs.

x +7x+10=x+6x +6x+4=0

2

2

-3+ 5 -3- 5√ √

- Multiply-Simplify

-Use the quadratic formula to find the answers

Examples

x +3x-10=92

x +3x-19=0

x=3.110

x=-3.110

-Multiply the logs together

-Cancel the logs and square 3

log(x-2)+log(x+5)=2log(3)2

-Simplify

-Use quadratic formula

2

log(x +3x-10)=log 3

Examples

You Try!1.)ln(x-3)+ln(x+4)=3ln(2)

2.)log (1-x)=13.)1\2ln(x+3)-ln(x)=04.)logx-(1\2)log(x+4)=15.)logx=46.)log (x-5)=-17.)4 =968.)4+log (7x)=10

x=-5, x=4x= -3x= ,x=103.85x=10,000x= 21\4x=5x=104.1433

4

1+ 13√2

1- 13√2

4

x2

Newton's Law of Cooling

T(t)-Tm=(To-Tm)e-Kt

Newton's Law of CoolingWhat Its Used For

• Newton’s Law of Cooling is used to measure the amount of heat lost over a period of time. This equation is often used in Occupations such as Crime scene investigation, where investigators examine bodies to see how long the person has been dead for.

• T(t)-Tm=(To-Tm)e-Kt T(t)=Temperature of object at time T Tm=Temperature of surrounding medium

(To-Tm)=Initial Temperature of the object K=Constant .25387 T=Time

Example

Lets say we (hypothetically) find Chris Kryshak “Accidentally” killed by someone. On the body we find a post-it note left by the killer (definitely not Eric Weber) That says the temperature on death was 37 degrees Celsius. Investigators determine that the current temperature of the body is 30degrees Celsius, and today is a warm day at 29 degrees Celsius. Using T(t)-Tm=(To-Tm)e-Kt

we can determine how long Chris has been dead.

Figuring Out What Time the Killer (Cough*Eric*Cough) killed Chris

• T(t)-Tm=(To-Tm)e-Kt Original Equation

• 30-29=(37)e.25387(T) Subtract 29 from 30

• 1=(37)e.25387(t) divide 1 by 37

• .02702703=e.25387(t) convert to Ln

• Ln.02702703=.25387t find Ln.02702703

• -3.610917803=.25387t Divide -3.610917803 by .25387

• T= -14.22349156*So, We have determined that Chris was killed by someone (who isn’t Eric) 14.22349156 hours ago. *This Number is a negative because it is looking at the time that has passed.