MODERATE EXERCISES-Logarithms

(1) Simplify.

1. ln
   
   
2. ln e^2x
   
   
3. e^{ln 3}

SOLUTION:

(a) We can write =x^1/2, so ln = ln x^1/2.
Then we can use the power property of logarithms (log_a x^r = r log_a x):
ln x^1/2 = (1/2)ln x . So the solution is (1/2)ln x, or (ln x)/2.

(b) Since log_a a^y = y, when using natural logarithms, the analogous statement would be: ln e^y = y. So we have:
ln e^2x = 2x .

(c) From another property of logarithms, we have e^{ln y} = y, since ln = log_e. So e^{ln 3} = 3.