A population of bacteria decreases according to the following model: $P(t)=20(0\mathrm{.}95{)}^t$ where $P(t)$ is the population in thousands and $t$ is the number of days since January $1^{st}(t=0$ represents January $1^{st})$

a$)$ How many bacteria are there on January $1^{st?}?$

b$)$ In how many days will the number of bacteria decrease to a half of the original amount?

Consider the transformed function $g(x)=-3(\frac{1}{5}{)}^x+2$ of the function $f(x)=(\frac{1}{5}{)}^x,$ Find the horizontal intercept algebraically and leave it in an exact form $($not decimal proximation$).$