Definition of LogarithmsWe recall from the last lesson that a logarithm is defined

as y = logbx if and only if By = x.

We will use this definition to solve equations involving logarithmic functions.

So…

3 = log7 x 73 = x

2 = logx 25 x2 = 25

y = log4 16 4y = 16

And…

62 = x 2 = log6 x

x3 = 8 3 = logx 8

8y = 64 y = log8 64

Rewriting Logarithms You can use the log properties to solve equations when the variable is contained in a logarithm.

1. Use the logarithm properties to rewrite as one log.

2. Rewrite the log into exponential form.

3. Solve • Raise 2 to the 5th

power• Distribute• Add 4 to both sides• Divide by the

coefficient

2 2

2

5

log 1 log 4 5

log 4 1 5

2 4 1

32 4 4

36 4

9

x

x

x

x

x

x

Equations with Natural LogsUse the same method when working with ln.

2

2

ln 1 1 3

ln 1 2

1

1

x

e

e

x

x

x

1.Isolate the ln

2.Rewrite in exponential form

Remember, natural logs have a base of e

3.Isolate variable

Application of LogarithmsIn 1906, San Francisco suffered a magnitude 7.8 (by many estimates) earthquake that caused unthinkable damage to the city.

To read more details about the quake go to:http://en.wikipedia.org/wiki/1906_San_Francisco_earthquake

The magnitude of an earthquake can be calculated using the

function y = log(1000x), where x represents the seismographic

reading 100 km from the center of the quake. What was the

Seismographic reading, in mm, for this earthquake?

Application of Logs, con’t1. Identify

variables

2. Sub in values3. Rewrite in

exponential form4. Isolate the

variable

10

7.8

7.8 log 1000

10 1000

63095734.45 1000

63095.73 mm

x

x

x

x

y= 7.8

y = log(1000x)

The seismographic reading 100 km from the center of the quake is ≈ 63,096 mm.

Additional Resourceshttp://math.usask.ca/emr/menu_exp.html

http://www.purplemath.com/modules/solvelog.htm

http://www.purplemath.com/modules/solvelog2.htm

http://www.wtamu.edu/academic/anns/mps/math/mathlab/col_algebra/col_alg_tut46_logeq.htm

http://www.phschool.com/webcodes10/index.cfm?fuseaction=home.gotoWebCode&wcprefix=age&wcsuffix=0805