Question

Algebra/Trig. I literally have no clue

Answer 1

Since the base of our logarithmic function is unknown, we can just call it:

$f(x)=\log_k(x)$

We are then given the point, (4,-2), so plugging in 4 for x and -2 for f(x) we get:

$-2=\log_{k}(4) \implies \\ -2=\frac{\log{4}}{\log{k}} \implies \\ -2\log{k}=\log{4} \implies \\ -2\log{k}=2\log{2} \implies \\ \log{k}=\frac{2\log{2}}{-2} \implies\\ \log{k}=-\log{2} \implies\\ k=10^{-\log{2}} \implies \\ k=\frac{1}{10^\log{2}}} \implies \\ k=\frac{1}{2}$

So our base is 1/2.