Key Terms
Example
Ln(x^2) = ln(2x+3) gives x = 3 or x = -1. Both check out here because x^2 is positive for either value.
Goal
Combine a sum or difference of logs into one single log. Order of operations: power rule first, then product rule, then
Order of operations
Quotient rule first, then product rule, then power rule.
Steps
1. Use log rules to condense each side to one log with the same base.
Solve
T = ln(5) / 2
Form
Y = A*e^(kt)
Definition
Log_b(S) = c means b^c = S
YES
Use one-to-one property; set arguments equal; solve; check.
NO
Condense one side first, then apply one of the above.
ALWAYS
Check solutions in the original equation before writing your answer.
Extraneous solution
A value produced algebraically that does not satisfy the original equation; usually caught by checking arguments of loga
Change-of-base formula
Converts any logarithm to a quotient of logs with a new base; used to evaluate non-standard bases on a calculator.
Power rule for logarithms
Log_b(M^n) = n * log_b(M); exponent moves in front as a multiplier.
Product rule for logarithms
Log_b(MN) = log_b(M) + log_b(N); log of a product becomes a sum.
Quotient rule for logarithms
Log_b(M/N) = log_b(M) - log_b(N); log of a quotient becomes a difference.