I am to solve $3+e^{2t}=7e^{2t}$. My textbook solution says it's $t=\ln\left(\frac{1}{\sqrt{2}}\right)=-\frac{1}{2}\ln(2)$ whereas I got $t=\frac{1}{2} \ln(\frac{1}{2})$
My working:
$$3+e^{2t}=7e^{2t}$$
$$3=6e^{2t}$$
$$\frac{1}{2}=e^{2t}$$
$$\ln\left(\frac{1}{2}\right)=2t$$
$$t=\frac{1}{2} \ln\left(\frac{1}{2}\right)$$
Where did I go wrong and how can I arrive at $t=\ln\left(\frac{1}{\sqrt{2}}\right)=-\frac{1}{2}\ln(2)$
Best Answer
Just note that $$t_{\text{yours}}=\frac{1}{2}\ln(\frac{1}{2})=\color{blue}{\frac{1}{2}\ln(2^{-1}) = \frac{1}{2}(-1)\ln(2)} = -\frac{1}{2}\ln(2) = t_{\text{book's}}$$