The Product Law

• add logarithms with the same base combining them into one logarithm and by multiplying their arguments
• log[b]x + log[b]y = log[b](xy)

The Quotient Law

• subtract logarithms with the same base combining them into one logarithm and by dividing their arguments
• log[b]x + log[b]y = log[b](x/y)

The Power Law

• if the argument in a logarithm is raised to an exponent, you can move the exponent so it becomes a coefficient of the logarithm
• log[b](x^n) = n*log[b]
• summary: you can convert exponents to coefficients and vise versa

Using Logarithm Laws to Expand Logarithms

• any variable in the numerator will become a positive logarithm (product law)
• any variable in the denominator will become a negative logarithm (quotient law)
• any exponent of an argument will become a coefficient (power law)

Using Logarithm Laws to Simplify Logarithms

• any coefficients will become exponents (power law)
• any positive logarithms will be in the numerator (product law)
• any negative logarithms will be in the denominator (quotient law)

Using Logarithm Laws to Evaluate Logarithms Example #1

Using Logarithm Laws to Evaluate Logarithms Example #2

Using Logarithm Laws to Evaluate Logarithms Example #3

Using Logarithm Laws to Evaluate Logarithms Example #4

Using Logarithm Laws to Evaluate Logarithms Example #5

Changing the Base of a Logarithm

• log[b]c = log[a]c / log[a]b
  • where "a" is any new base
  • the argument goes in the numerator
  • the base goes in the denominator

Evaluating Logarithms Using a Graphing Calculator

• change the logarithm to base 10