Question

precalculus ii 92.1230
quick quiz 1
calculate:
log₄(1/16)=
ln√e=

Explanation:

Step1: Recall log - rule for $\log_4\frac{1}{16}$

Let $y = \log_4\frac{1}{16}$. By the definition of logarithms, if $y=\log_a x$, then $a^y=x$. So we need to find $y$ such that $4^y=\frac{1}{16}$. Since $4 = 2^2$ and $\frac{1}{16}=2^{- 4}$, then $(2^2)^y=2^{-4}$, which simplifies to $2^{2y}=2^{-4}$. So $2y=-4$, and $y = - 2$.

Step2: Recall log - rule for $\ln\sqrt{e}$

Let $z=\ln\sqrt{e}$. Since $\ln$ is the natural logarithm with base $e$ (i.e., $\ln x=\log_e x$) and $\sqrt{e}=e^{\frac{1}{2}}$. By the definition of logarithms, if $z = \log_e e^{\frac{1}{2}}$, then $e^z=e^{\frac{1}{2}}$. So $z=\frac{1}{2}$.

Answer:

$\log_4\frac{1}{16}=-2$, $\ln\sqrt{e}=\frac{1}{2}$