[tex]\bf \begin{array}{llll} \textit{logarithm of factors} \\\\ \log_a(xy)\implies \log_a(x)+\log_a(y) \end{array} ~\hspace{4em} \begin{array}{llll} \textit{Logarithm of rationals} \\\\ \log_a\left( \frac{x}{y}\right)\implies \log_a(x)-\log_a(y) \end{array}[/tex]
[tex]\bf \begin{array}{llll} \textit{Logarithm of exponentials} \\\\ \log_a\left( x^b \right)\implies b\cdot \log_a(x) \end{array} ~\hspace{7em} \begin{array}{llll} \textit{Logarithm Cancellation Rules} \\\\ \stackrel{\textit{let's use this one}}{log_a a^x = x}\qquad \quad a^{log_a x}=x \end{array} \\\\[-0.35em] \rule{34em}{0.25pt}\\\\ 900=300e^{0.03t}\implies \log_e(900)=\log_e(300e^{0.03t})[/tex]
[tex]\bf \log_e(900)=\log_e(300)+\log_e(e^{0.03t})\implies \ln(900)=\ln(300)+0.03t\cdot \ln(e) \\\\\\ \ln(900)-\ln(300)=0.03t\implies \ln\left( \cfrac{900}{300} \right)=0.03t\implies \ln(3)=0.03t[/tex]
