Exponential and logarithmic functions

Compute $2^{-3} \cdot 2^{5}$.

Compute $\log_3 81$.

Compute $16^{3/4}$.

For $x > 0$, which expression equals $\log_2(8x)$?

Solve $5 \cdot 3^x = 200$.

A hospital receives $160$ mg of iodine-131, which has a half-life of $8$ days. How many milligrams remain after $24$ days?

In the proof that $\log_2 3$ is irrational we reach $2^m = 3^n$ with whole numbers $m \ge 1$ and $n \ge 1$. Which of these facts would finish the proof? Choose all that apply.

Let b : ℝ with 0 < b and b ≠ 1. Which Lean statement says that $x \mapsto b^x$ is a bijection from $\mathbb{R}$ onto $(0, \infty)$?

Savings grow by a rate $r$ per year (for example $r = 0.05$ for $5\%$). Write years_to_double(r), which returns the smallest whole number of years $n$ with $(1 + r)^n \ge 2$, and doubling_time(r), which returns the exact doubling time $\dfrac{\ln 2}{\ln(1 + r)}$ using math.log.