已知 $9^{12} = 27^{x}$，求 $x$。 答案 $x =$ ... 某国拥有汽车的人数每年增长 $p\%$，从 2020 年的 5000 人增加到 2024 年的 6500 人。 求 $p$ 的值。 答案 $p =$ ...

1. Express both powers using base 3.

Since $9=3^2$ and $27=3^3$,
$9^{12}=27^x$
$(3^2)^{12}=(3^3)^x$

2. Simplify the indices and equate them.

$3^{24}=3^{3x}$
Since the bases are equal,
$24=3x$
$x=8$

1. Form an equation using the compound increase multiplier.

From 2020 to 2024, there are $4$ yearly increases.

If the yearly percentage increase is $p\%$, the multiplier each year is $1+\dfrac{p}{100}$.

$5000\left(1+\frac{p}{100}\right)^4=6500$

2. Solve for the yearly percentage increase.

$\left(1+\frac{p}{100}\right)^4=\frac{6500}{5000}=1.3$

$1+\frac{p}{100}=\sqrt[4]{1.3}$

$p=100\left(\sqrt[4]{1.3}-1\right)$

$p=6.779\ldots$

So, to 3 significant figures,

$p=6.78$