Which operation can be used to eliminate the natural logarithm ($\ln$) from both sides of an equation such as $\ln \left(x\right)=3$?

Computer Explorations

Use a CAS to perform the following steps in Exercises 55–62.

a. Plot the equation with the implicit plotter of a CAS. Check to see that the given point P satisfies the equation.

2y² + (xy)¹/³ = x² + 2, P(1,1)

Which of the following best describes the function $Ei\left(x\right)$?

Which of the following transformations results in a vertical stretch of the exponential decay function $f\left(x\right)=e^{-x}$?

Given the following table of values for an exponential function $f\left(x\right)$:

Suppose a quantity described by the function y(t) = y₀eᵏᵗ, where t is measured in years, has a doubling time of 20 years. Find the rate constant k.

Geometric means A quantity grows exponentially according to y(t) = y₀eᵏᵗ. What is the relationship among m, n, and p such that y(p) = √(y(m)y(n))?

"General relative growth rates Define the relative growth rate of the function f over the time interval T to be the relative change in f over an interval of length T:

R_T = [f(t + T) − f(t)] / f(t)

Show that for the exponential function y(t) = y₀ e^{kt}, the relative growth rate R_T, for fixed T, is constant for all t."