To rewrite the function \( f(x) = \log_{11}(x - 9) \) in exponential form, we start by recalling the relationship between logarithms and exponents. The logarithmic equation \( \log_{b}(a) = c \) is equivalent to the exponential form \( b^c = a \).
In this case, we can set the function equal to \( y \): \[ y = \log_{11}(x - 9) \]
This implies: \[ 11^y = x - 9 \]
To rearrange this into a form that isolates \( x \), we add 9 to both sides: \[ x = 11^y + 9 \]
Now we can analyze the students' responses:
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Student #1: \( y11 + 9 = x \) - This is not correctly written since \( y \cdot 11 + 9 \) does not match the form we derived.
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Student #2: \( 11y + 9 = x \) - This is also not correct; it seems to imply multiplication rather than an exponent.
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Student #3: \( 11^y + 9 = x \) - This is the correct form we derived.
Thus, Student #3 correctly rewrote the function \( f(x) = \log_{11}(x - 9) \) in exponential form.