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 4 - 2020 - Paper 1

Rearranging the equation for $p$ gives:

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

Thus,

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

and dividing by 3 gives:

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

Using a calculator to find the values:

• Calculate $\text{log}(5) \approx 0.6990$
• Calculate $\text{log}(4) \approx 0.6021$

Substituting these values into the equation:

$p = \frac{1}{3} \left( \frac{210 \times 0.6990}{0.6021} + 1 \right) \approx 27.1.$

Therefore, to one decimal place, the value of $p$ is approximately:

$p \approx 27.1.$