Solve $\log_x 3 - \log_3 3 = 2.$ - Scottish Highers Maths - Question 3 - 2023

To start solving the equation, we can use the property of logarithms that states $\log_a b - \log_a c = \log_a \left( \frac{b}{c} \right)$. Thus:

$\log_x 3 - \log_3 3 = 2 \implies \log_x 3 = 2 + \log_3 3$

Next, we know that $\log_3 3 = 1$. Therefore:

$\log_x 3 = 2 + 1 = 3$

Now, we can convert the logarithmic equation into its exponential form:

$x^3 = 3$

To find $x$, we take the cube root of both sides:

$x = 3^{\frac{1}{3}} = \sqrt[3]{3}$