PROPERTIES OF LOGARITHMS

Property 1

If x and y are positive real numbers, b is positive and not equal to 1, then

log_bxy = log_bx + log_by Examples:

log₄35 = log₄(7*5) = log₄7 + log₄5
log₃21 = log₃(7*3) = log₃7 + log₃3
ln18 = ln(2*3*3) = ln2 + ln3 + ln3

Property 2

If x and y are positive real numbers, then

log_bx/y = log_bx - log_by Examples:

log₁₀10/2 = log₁₀10 - log₁₀2
log_e50/5 = log_e50 - log_e5
log₁₀100/3 = log₁₀100 - log₁₀3
log₁₀₀10/2 = log₁₀₀10 - log₁₀2

Property 3

If x and y are positive real numbers, then

log_bx^p = p log_bx Examples:

log5³ = 3log5
log₂16⁵ = 5log₂16

Property 4

If x and y are positive real numbers,then

log_b1 = 0

Property 5

If x and y are positive real numbers,then

log_bb = 1

Property 6

If x and y are positive real numbers,then

log_bbⁿ = n