2. Definition of Logarithms We recall from the last lesson that a logarithm is defined as y = log b x if and only if B y = x. We will use this definition to solve equations involving logarithmic functions. So… 3 = log 7 x 7 3 = x 2 = log x 25 x 2 = 25 y = log 4 16 4 y = 16 And… 6 2 = x 2 = log 6 x x 3 = 83 = log x 8 8 y = 64y = log 8 64

3. 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 5 th power Distribute Add 4 to both sides Divide by the coefficient

4. Equations with Natural Logs Use the same method when working with ln. 1.Isolate the ln 2.Rewrite in exponential form Remember, natural logs have a base of e 3.Isolate variable

5. 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?

6. Application of Logs, con’t 1. Identify variables 2. Sub in values 3. Rewrite in exponential form 4. Isolate the variable y= 7.8 y = log(1000x) The seismographic reading 100 km from the center of the quake is ≈ 63,096 mm.

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