By taking logarithms of both sides, solve the equation $$4^{3p - 1} = 5^{210}$$ giving the value of $p$ to one decimal place. - Edexcel - A-Level Maths Pure - Question 3 - 2020 - Paper 1

Now, isolate the term containing $p$:

$3p - 1 = \frac{210 \cdot \text{log}(5)}{\text{log}(4)}$

Add 1 to both sides:

$3p = \frac{210 \cdot \text{log}(5)}{\text{log}(4)} + 1$

Finally, divide by 3 to solve for $p$:

$p = \frac{1}{3} \left( \frac{210 \cdot \text{log}(5)}{\text{log}(4)} + 1 \right)$

Now, using a calculator,

1. Calculate $\text{log}(5)$ and $\text{log}(4)$
2. Substitute these values into the expression for $p$:

Calculating gives approximately $p \approx 81.6$.

Thus, the value of $p$ to one decimal place is $81.6$.