how do you write a logarithm as the sum or difference of logarithms.

Question

example 1: ln(e³x²/³√y)

example 2: log (100/(x²-y²)⁵

Answer 1

lna^b= blna

ln(ab)= lna+lnb

ln(a/b)= lna- lnb

lne =1

apply the above laws of logs

ln(e^3x^2/y^(1/3)

=3lne +2lnx - (1/3)lny

= 3 +2lnx +(1/3)lny

log(100/(x^2-y^2)^5)

=log100-5log(x^2-y^2)

=log(10^2) - 5log((x+y)(x-y))

=2log10 -5log(x+y)- 5log(x-y)

= 2 - 5log(x+y) -5log(x-y)

Answer 2

example 1:

ln(e³x²/³√y) = ln(e³) + ln(x²) - ln ( y^1/3)

= 3 + 2ln(x) -ln(y)/3

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example 2:

log (100/(x²-y²)⁵) = log(100) - 5log (x² - y2)

= 2 - 5log[(x+y)*(x-y)]

= 2 - 5log(x+y) -5log(x-y)

Note: there is a problem with unbalanced parentheses above in your copy of the problem. Can you check to see if I placed it where it should have been?