Solve 5^(4x – 1) = 61

To address the equation 5^(4x – 1) = 61, one must first isolate the variable x. This can be achieved by applying logarithmic functions to both sides of the equation. By taking the logarithm base 5 of both sides, we can transform the equation into a more manageable form. This step allows us to express the exponent in terms of logarithms, leading to the equation (4x – 1) = log_5(61). 

Next, we can solve for the variable x by rearranging the equation. This involves adding 1 to both sides and then dividing by 4. The resulting expression will yield x = (log_5(61) + 1) / 4. To further simplify the calculation, one may convert the logarithm to a more familiar base, such as base 10 or the natural logarithm, using the change of base formula. This will facilitate the computation of the logarithmic value, allowing for a precise determination of x.

In the context of preparing for examinations, such as during the Easter Half Term A Level Maths Revision, mastering the techniques for solving exponential equations is crucial. Understanding how to manipulate logarithms and exponents not only aids in solving this particular equation but also enhances overall mathematical proficiency. This foundational knowledge is essential for tackling a variety of problems encountered in advanced mathematics.