Given that $9^{12} = 27^{x}$, find $x$. Answer $x =$ ... The number of people owning cars in a country increased by $p\%$ every year, from 5000 in the year 2020, to 6500 in the year 2024. Find the value of $p$. Answer $p =$ ...

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,

$p=6.78$