Question

What is t. with explanation

Answer 1

`\bf \textit{Logarithm Cancellation Rules} \\\\ \stackrel{\stackrel{\textit{we'll use this one}}{\downarrow }}{log_a a^x = x}\qquad \qquad a^(log_a x)=x \\\\[-0.35em] \rule{34em}{0.25pt}\\\\ 12\cdot 6^(t-1)=6\cdot e^(3t-1) \\\\[-0.35em] ~\dotfill\\\\ 12\cdot 6^t\cdot 6^(-1)=6\cdot e^(3t)\cdot e^(-1)\implies 12\cdot \cfrac{6^t}{6}=6\cdot \cfrac{e^(3t)}{e}\implies 2\cdot 6^t=6\cdot \cfrac{e^(3t)}{e}`

`\bf \cfrac{2\cdot 6^t}{6}=\cfrac{e^(3t)}{e}\implies \cfrac{6^t}{3}=\cfrac{e^(3t)}{e}\implies e6^t=3e^(3t)\implies ln(e6^t)=ln(3e^(3t)) \\\\\\ ln(e)+ln(6^t)=ln(3)+ln(e^(3t))\implies 1+ln(6^t)=ln(3)+3t\cdot  ln(e) \\\\\\ 1+t\cdot  ln(6)=ln(3)+3t\implies t\cdot ln(6)-3t=ln(3)-1 \\\\\\ t[ln(6)-3]=ln(3)-1\implies t=\cfrac{ln(3)-1}{ln(6)-3}\implies t\approx -0.08161643824769`