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Consider the function 𝑓(𝑥) = log₂ (3𝑥 − 1). If 𝑓(𝑎) = 3, find the value of 𝑎.

Consider the function 𝑓 of 𝑥 is equal to log base two of three 𝑥 minus one. If 𝑓 of 𝑎 is equal to three, find the value of 𝑎.

We are told that 𝑓 of 𝑎 is equal to three, so we can begin by substituting these values into the function 𝑓 of 𝑥. This gives us log base two of three 𝑎 minus one is equal to three. We recall that logarithmic functions are the inverses of exponential functions. This means that if log base 𝑎 of 𝑦 is equal to 𝑥, then 𝑎 to the power of 𝑥 is equal to 𝑦. In this question, the base 𝑎 is equal to two, the variable 𝑦 is equal to three 𝑎 minus one, and the variable 𝑥 is equal to three. Two cubed is therefore equal to three 𝑎 minus one.

We know that two cubed is equal to eight. We can then add one to both sides of this equation so that three 𝑎 is equal to nine. Dividing both sides of this equation by three gives us 𝑎 is equal to three. If the function 𝑓 of 𝑥 is equal to log base two of three 𝑥 minus one and 𝑓 of 𝑎 is equal to three, then the value of 𝑎 is three. We could check this answer on the calculator by substituting our value back in to the original function.

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