0. Functions

Which operation can be used to eliminate a natural logarithm, such as $\ln \left(x\right)$, from an equation?

Convert the following logarithmic expression to its equivalent exponential form.

$\log_4x=5$

Convert the following logarithmic expression to its equivalent exponential form.

$x=\log9$

Convert the following exponential expression to its equivalent logarithmic form.

$3^{x}=7$

Convert the following exponential expression to its equivalent logarithmic form.

$e^9=x+3$

{Use of Tech} Let f(x) = ln((x+1)/(x-1)) and g(x) = ln ((x+1)/(x-1)).

b. Sketch graphs of f and g to show that these functions do not differ by a constant.

Convert the following expressions to the indicated base.

$2^x$ using base e

Solving equations Solve the following equations.

log₈ x = 1/3

Find the inverse $f^{-1}\left(x\right)$ of each function (on the given interval, if specified).

$f\left(x\right)=e^{2x+6}$

What are the domain and range of ln x?

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.

Falling body When an object falling from rest encounters air resistance proportional to the square of its velocity, the distance it falls (in meters) after t seconds is given by d(t) = (m/k) ln (cosh (√(kg/m) t)), where m is the mass of the object in kilograms, g = 9.8 m/s² is the acceleration due to gravity, and k is a physical constant.

a. A BASE jumper (m = 75 kg) leaps from a tall cliff and performs a ten-second delay (she free-falls for 10 s and then opens her chute). How far does she fall in 10 s? Assume k = 0.2.