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Question Video: Evaluating a Logarithmic Function at a Given Point Mathematics • 10th Grade

Consider the function 𝑓(π‘₯) = logβ‚‚ (3π‘₯ βˆ’ 1). If 𝑓(π‘Ž) = 3, find the value of π‘Ž.

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Video Transcript

Consider the function 𝑓 of π‘₯ is equal to log base two of three π‘₯ minus one. If 𝑓 of π‘Ž is equal to three, find the value of π‘Ž.

We are told that 𝑓 of π‘Ž is equal to three, so we can begin by substituting these values into the function 𝑓 of π‘₯. This gives us log base two of three π‘Ž minus one is equal to three. We recall that logarithmic functions are the inverses of exponential functions. This means that if log base π‘Ž of 𝑦 is equal to π‘₯, then π‘Ž to the power of π‘₯ is equal to 𝑦. In this question, the base π‘Ž is equal to two, the variable 𝑦 is equal to three π‘Ž minus one, and the variable π‘₯ is equal to three. Two cubed is therefore equal to three π‘Ž 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 π‘Ž is equal to nine. Dividing both sides of this equation by three gives us π‘Ž is equal to three. If the function 𝑓 of π‘₯ is equal to log base two of three π‘₯ minus one and 𝑓 of π‘Ž is equal to three, then the value of π‘Ž is three. We could check this answer on the calculator by substituting our value back in to the original function.

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