Section 10.2: Logarithmic Equations

10.2 Outline

1. Fundamental properties
   1. logarithmic equation
   2. Grant’s tomb properties
      1. x is the exponent on a base b that gives bx.
      2. log_bx is the exponent on a base b that gives x.
2. Logarithmic equations
   1. log of both sides theorem
   2. four types
      1. the unknown is the logarithm
      2. the unknown is the base
      3. the logarithm of an unknown is equal to a number
      4. the logarithm of an unknown is equal to the logarithm of a number
3. Laws of logarithms
   1. additive law
   2. subtractive law
   3. multiplicative law

10.2 Essential Ideas

Fundamental properties of logarithms (Grant’s tomb properties):

log_bb^x=x
b^xlogb^x = x

logarithmic equation is an equation for which there is a logarithm on one or both sides.  The key to solving logarithmic equations is the log of both sides theorem:

If A, B, and b are positive real numbers (with b not equal to 1), then log _bA = log_bB is equivalent to A = B.

Laws of Logarithms

If A, B, and b are positive numbers, p is any real numbers and b is not equal to 1:

Addition Law:

log_b(AB) = log_bA + log_bB

Subtraction Law:

log_b(A/B) = log_bA – log_bB

Multiplication Law:

log_bA^p = plog_bA