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

Next, isolate $p$ by dividing both sides by $\log 4$:

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

Then, add 1 to both sides:

$3p = \frac{210 \log 5}{\log 4} + 1$

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

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

Now, calculating this with the values of $\log 5 \approx 0.6990$ and $\log 4 \approx 0.6021$:

• First, calculate $\frac{210 \times 0.6990}{0.6021} \approx 245.8$
• Then, add 1: $245.8 + 1 \approx 246.8$
• Finally, divide by 3: $\frac{246.8}{3} \approx 82.3$

Thus, the final value of $p \approx 82.3$, rounded to one decimal place.