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\Huge{
Logarithmic Properties}
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\noindent
While working in a group make sure you:
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\item Expect to make mistakes but be sure to reflect/learn from them!
\item Are civil and are aware of your impact on others.
\item Assume and engage with the strongest argument while assuming best intent.
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Let $b>0$, $b\not=0$ and let $v$ and $w$ be positive real numbers.  Let $k$ be a real number.
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\item $\log_b(b^x)=x$ \hspace{1.0in} $\displaystyle{b^{\log_b(x)}=x}$
\item $\log_b(vw)=\log_b(v)+\log_b(w)$
\item $\displaystyle{\log_b\left(\frac{v}{w}\right)=\log_b(v)-\log_b(w)}$
\item $\log_b(v^k)=k\log_b(v)$
\item $\displaystyle{\log_b(x)=\frac{\log (x)}{\log(b)}}$ \hspace{1.0in} $\displaystyle{\log_b(x)=\frac{\ln (x)}{\ln(b)}}$ 
\end{enumerate}


\begin{enumerate}




\item Write the expressions as a single logarithm:

$\ln(x)-\ln(y)+3\ln(z)$ \hspace{2in} $\frac{1}{3}\log (2x+1)-2\log(x^4-x^2-1)$
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\item Expand the expressions:

$\log_2(2xy^3)$ \hspace{3in} $\displaystyle{\log \left(\frac{a\sqrt{c}}{b^4}\right)}$

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\item Given $\ln(x)=5$ and $\ln(y)=7.5$, find $\displaystyle{\ln\left(\frac{x^2}{y}\right)}$.
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\item Find all $x$ that satisfy:
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\item $2000e^{.05x}=10,000$
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\item $\log (x-16)=2- \log(x-1)$
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\item $\log(x+1)+\log(x-1)=\log 1$
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\item $\displaystyle{\frac{10}{1+e^{-x}}=2}$
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\item $\displaystyle{7^{\frac{x}{3}\ln 5}=9}$
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\item $\log_2(\log_3(x))=4$
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\end{enumerate}
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