Final answer:
The correct first step in solving the equation 5ⁿ = 21 is to apply logarithms to both sides to bring down the exponent, which is option C: log 5ⁿ = log 21. This approach 'inverts' the exponential function, making it possible to solve for x. Therefore, the correct answer is option C. log 5ˣ = log 21.
Step-by-step explanation:
The first step in solving the exponential equation 5ⁿ = 21 is to find a way to deal with the exponent. While options A, B, and D are not suitable for isolating the variable x, option C effectively allows us to 'invert' the exponential function by applying logarithms. The correct first step is option C: log 5ⁿ = log 21.
Logarithms are particularly useful for solving equations where the variable is an exponent, because applying a logarithm to both sides of an equation allows us to bring the variable down from the exponent. This principle is based on the relationship between exponents and logarithms: if we know that aˣ = b, then it must also be true that
loga b = c.
By applying the logarithm with the same base as the base of the exponential function, we essentially 'undo' the exponential, leaving us with x, which can then be easily isolated and solved for.