Precalculus 3.3 notes: Properties of Logarithms

3.3 Properties of Logarithms

Change of Base:
Let a, b, and x be positive real numbers such that a≠1 and b≠1. Then log_ax can be converted to a different base using any of the following formulas.

Base b

log_ax=(log_bx)/(log_ba)

Base 10

log_ax=(log₁₀x)/(log₁₀a)

Base e

log_ax = (ln x)/(ln a)

Examples:

log₇4 = .7124143742
log₁₅1460 = 2.690567447

log_(1/3)(0.015)=3.822736302

Properties of Logarithms:
Let a be a positive number such that a≠1, and let n be a real number. If u and v are positive real numbers, the following properties are true.

1. log_a(uv) = log_au + log_av

2. ln(uv) = ln u + ln v

3. log_a (u/v) = log_au - log_av

4. ln (u/v) = ln u - ln v

5. log_auⁿ = n log _au

6. ln uⁿ = n ln u

Examples: Expand or condense:

1. ln (x⁴ y½)/(y²z³) = 4ln x + 1/2 ln y - 2 ln y - 3 ln z = 4 ln x - 3/2 ln y - 3 ln z

2. ln (x⁴(x²+1))^¼= (1/4)( 4 ln x + ln (x² + 1))

3. 4(lnz + ln(z+5)) - 2ln(z-5) + 3ln(x-2) =