i get a negative number every time i try to solve...
Let's do it this way then.
Rewrite $\displaystyle e^{\frac{ln(3/5)t}{5}}=(3/5)^{\frac{t}{5}}$
$\displaystyle 50=100(3/5)^{\frac{t}{5}}$
$\displaystyle 1/2=(3/5)^{\frac{t}{5}}$
$\displaystyle ln(1/2)=\frac{t}{5}ln(3/5)$
$\displaystyle t=\frac{5ln(1/2)}{ln(3/5)}=6.78$
Hello,
Of base 10 ? o.O
Put it in the exponential logarithm... $\displaystyle \log_x (y)=\frac{\ln(y)}{\ln(x)}$
---> $\displaystyle \frac{\ln x}{\ln 3}-\frac{\ln 7}{\ln 4}=\frac{\ln 5}{\ln 7}$
--> $\displaystyle \ln x-\frac{\ln 7 \cdot \ln 3}{\ln 4}=\frac{\ln 5 \cdot \ln 3}{\ln 7}$
$\displaystyle \ln x=\frac{\ln 5 \cdot \ln 3}{\ln 7}+\frac{\ln 7 \cdot \ln 3}{\ln 4}$
---> $\displaystyle x=exp\left(\frac{\ln 5 \cdot \ln 3}{\ln 7}+\frac{\ln 7 \cdot \ln 3}{\ln 4}\right) \approx 11.59699220980753$
For the last one
$\displaystyle a^{bc}=(a^b)^c$
So $\displaystyle e^{\ln(\frac{3}{5})\frac{t}{5}}=(e^{\ln(\frac{3}{5 })})^{\frac{t}{5}}=\bigg(\frac{3}{5}\bigg)^{\frac{ t}{5}}$
and no multiplying by it does nothign
raising e to each side power does
for example $\displaystyle \ln(45x-220)=\ln(5+17x)$
If you multiplied both sides by e would do nothing
but raising e to eachside would give
$\displaystyle e^{\ln*45x-220)}=e^{\ln(5+17x)}\Rightarrow{45x-220=5+17}$