What is the inverse function of ln x, and what are its domain and range?

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample. Assume x > 0 and y > 0.

b. ln 0 = 1

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample. Assume x > 0 and y > 0.

c. ln (x + y) = ln x + ln y

Logarithm properties Use the integral definition of the natural logarithm to prove that ln(x/y) = ln x - ln y.

ln x is unbounded Use the following argument to show that lim (x → ∞) ln x = ∞ and lim (x → 0⁺) ln x = −∞.

c. Show that ln 2ⁿ > n/2 and ln 2^(−n) < −n/2.

Convert the following logarithmic expression to its equivalent exponential form.

$\(\log\)_4x=5$

Convert the following logarithmic expression to its equivalent exponential form.

$x=\(\log\)9$