Solve the inequality for x. (enter your answers using interval notation.) ln x < 0

ok. let's solve ln x <0

solve ln x = 0

raise e to both sides

e^lnx = e^0
x=1

so we make a number line:
we add this root and any place that ln is undefined. you can't take the ln of anything less than or equal to 0, so we also put 0.

-undefneg...
01

we test a number bigger than 1 and a number less than 1.

since e>1, we can use it, and ln e = ln(e^1)= 1 > 0

since 0<1/e < 1, we can use it, and ln(1/e) = ln(e^-1) = -1 < 0

so ln x < 0 when 0

now lets try e^x > 6.

solve e^x = 6.

take ln of both sides.

x = ln 6.

< 6>6...
ln 6
e^x is always defined so we test a number bigger than ln 6 and a number less than ln 6.

0 < ln 6, so we can use it and e^0 = 1 < 6.

ln 7 > ln 6, since ln is always increasing. so we can use ln7, and e^ln 7 = 7 > 6.

so e^x > 6 for x>ln 6 or (ln 6, infinity).

Hope it helps