Simplify

a) 4ln2 + 2ln3

b) 0.5ln9 - ln2

c) 2ln5 - 1

d) 0.25ln81

We will use the following logarithm rules:

ln (a) + ln (b) = ln (ab)

ln (a) - ln (b) = ln (a / b)

aln (b) = ln (b) ᵃ

a) 4ln (2) + 2ln (3)

= ln (2) ⁴ + ln (3) ²

= ln (16 x 9)

= ln (144)

= 2ln (12)

b) 0.5ln (9) - ln (2)

= ln (√9) - ln (2)

= ln (3) - ln (2)

= ln (3 / 2)

c) 2ln (5) - 1

= ln (5) ² - ln (e)

= ln (25 / e)

d) 0.25ln (81)

= ln (81) ⁰.²⁵

= ln (3)

D) 0.25ln81 is in simplest form