Chapter 12.5, Problem 1PE

a. As x becomes increasingly large, the value of ${\left(1+\frac{1}{x}\right)}^x$ approaches ___________ $\approx 2.71828$.

b. The function $f\left(x\right)=e^x$ is the exponential function base ___________.

c. The logarithmic function base e is called the ___________ logarithmic function and is denoted by $y=$ ___________.

d. If x is a positive real number, then $\text{ln}1=$ _____, $\ln e=$ _____, $\ln e^p=$ _____, and $e^{\ln x}=$ _____.

e. If x and y are positive real numbers, then $\ln \left(xy\right)=$ _____ and $\ln \left(\frac{x}{y}\right)=$ _____.

f. If x is a positive real number, then for any real number p, $\ln x^p$ can be written as _____.

g. The change-of-base formula states that where a is a positive real number and $a\ne 1$.