simplifying logs?

Question

ahh, I have to put these into condensed form and simplify

1)2lne+4lne^2-2lne
2)2log0.1+4log0.01-2log0.001

the first one I'm clueless, but I got 1/1000 for the second one...

ahhh THANKS FOR HELPING

Answer 1

1) ln means "log base e." ln(e) = 1, and ln(e^x) = x.

2ln(e) + 4ln(e^2) - 2ln(e) = 2 + 4*2 - 2 = 8

2) log normally means "log base 10." log(0.01) = -2 because 0.01 = 10^(-2).

2log(0.1) + 4log(0.01) - 2log(0.001) = 2(-1) + 4(-2) - 2(-3) = -2 + (-8) - (-6) = -4

Answer 2

log x to power y= y log x --> log x^y= y log x
ln e = log e to base e = 1
log x = log x to base 10
log 10 to base 10 = 1
log 1 to base anything= 0


1)2ln e + 4ln e^2 - 2ln e
= 2 (1) + 4*2 ln e - 2 (1)
= 2 + 8 - 2 = 8 Ans

2) 2log0.1+4log0.01-2log0.001
= 2 log (10)^-1 + 4 log (10)^-2 - 2 log (10)^-3
= 2 (-1) log 10 + 4 (-2) log10 - 2 (-3) log 10
= -2 log 10 - 8 log 10 + 6log 10
= - 4 log 10
= -4 (1)
= - 4 Ans
Anti log (-4) = (10)^-4 = 1/ 10000