Chapter 12.4, Problem 1PE

a. Fill in the blanks to complete the basic properties of logarithms.

${\log }_{b}b=$ ________, ${\log }_{b}1=$ ________, ${\log }_b{b}^x=$ ________, and $b^{{\log }_{b}x}=$ ________.

b. If b, x, and y are positive real numbers and $b\ne 1,$ then ${\log }_b\left(xy\right)=$ ________, and ${\log }_b\left(\frac{x}{y}\right)=$ ________.

c. If b and x are positive real numbers and $b\ne 1$, then for any real number p, ${\log }_b^{}{x}^p$ can be written as ________.

d. Determine if the statement is true or false: ${\log }_b\left(xy\right)=\left({\log }_{b}x\right)\left({\log }_{b}y\right)$

Use the expression ${\log }_2(4\cdot 8)$ to help you answer.

e. Determine if the statement is true or false: ${\log }_b\left(\frac{x}{y}\right)=\frac{{\log }_{b}x}{{\log }_{b}y}$

Use the expression ${\log }_3\left(\frac{27}{9}\right)$ to help you answer.

f. Determine if the statement is true or false: ${\log }_b{\left(x\right)}^p={\left({\log }_{b}x\right)}^p$

Use the expression $\log {\left(1000\right)}^2$ to help you answer.