Definition of Logarithms
We recall from the last lesson that a logarithm is defined
as y = logbx if and only if By = x.
So…
And…
3 = log7 x
73 = x
62 = x
2 = log6 x
2 = logx 25
x2 = 25
x3 = 8
3 = logx 8
y = log4 16
4y = 16
8y = 64
y = log8 64
We will use this definition to solve equations involving logarithmic functions.
Rewriting Logarithms
You can use the log properties to solve equations when
the variable is contained in a logarithm.
Use the logarithm
properties to rewrite as one
log.
2. Rewrite the log into
exponential form.
3. Solve
1.
•
•
•
•
5th
Raise 2 to the
power
Distribute
Add 4 to both sides
Divide by the coefficient
log 2  x  1  log 2 4  5
log 2 4  x  1  5
2  4  x  1
5
32  4 x  4
36  4 x
x9
Equations with Natural Logs
Use the same method when working with ln.
1. Isolate the ln
ln  x  1  1  3
2. Rewrite in exponential
form
ln  x  1  2
Remember, natural logs have a base of e
3. Isolate variable
e  x 1
2
x  e 1
2
Application of Logarithms
In 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’t
1.
y= 7.8
Identify variables
2. Sub in values
3. Rewrite in exponential
form
4. Isolate the variable
y = log(1000x)
7.8  log10 1000 x 
10
7.8
 1000 x
63095734.45  1000 x
x  63095.73 mm
The seismographic reading 100 km from the center of the quake is ≈ 63,096 mm.
Additional Resources
 http://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/mathla
b/col_algebra/col_alg_tut46_logeq.htm
 http://www.phschool.com/webcodes10/index.cfm?fuseacti
on=home.gotoWebCode&wcprefix=age&wcsuffix=0805