Properties of Logarithms By: Jennifer Garcia & Roslynn Martinez

Basic Properties of Logarithms Most used properties

Using the Log Function for Solutions Consider solving Previously used algebraic techniques (add to, multiply both sides) not helpful Consider taking the log of both sides and using properties of logarithms

Try It Out Consider solution of 1.7(2.1) 3x = 2(4.5) x Steps Take log of both sides Change exponents inside log to coefficients outside Isolate instances of the variable Solve for variable

Natural Logarithms We have used base of 10 for logs Another commonly used base for logs is e e is an irrational number (as is ) e has other interesting properties Later to be discovered in calculus

Use Properties for Solving Exponential Equations Given Take log of both sides Use exponent property Solve for what was the exponent Note this is not the same as log 1.04 – log 3

Misconceptions log (a+b) NOT the same as log a + log b log (a-b) NOT the same as log a – log b log (a * b) NOT the same as (log a)(log b) log (a/b) NOT the same as (log a)/(log b) log (1/a) NOT the same as 1/(log a)

Usefulness of Logarithms Logarithms useful in measuring quantities which vary widely Acidity (pH) of a solution Sound (decibels) Earthquakes (Richter scale)

Example of Logarithms and Orders of Magnitude: Consider increase of CD's on campus since 1990 Suppose there were 1000 on campus in 1990 Now there are 100,000 on campus The log of the ratio is the change in the order of magnitude

Change of Base Formula We have used base 10 and base e What about base of another number  log 2 17 = ? Use formula Use base 10 or base e which calculator can do for you

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