Question - Logarithmic Equation Problem
Solution:
\[
\text{原方程式為：} \log_7(7^{2y} - 18) = y + 1
\]
\[
\text{假設} \log_7(7^{2y} - 18) = a \text{，則} 7^a = 7^{2y} - 18
\]
\[
\text{原方程式可寫作：} 7^{y + 1} = 7^{2y} - 18
\]
\[
\text{注意到} 7^{y+1} = 7^y \cdot 7 \text{，帶回方程式中：}
\]
\[
7^y \cdot 7 = 7^{2y} - 18
\]
\[
\text{令} x = 7^y \text{，則上式轉變為：}
\]
\[
7x = x^2 - 18
\]
\[
\text{此為一元二次方程，整理得：}
\]
\[
x^2 - 7x - 18 = 0
\]
\[
\text{解此一元二次方程得：}
\]
\[
(x - 9)(x + 2) = 0
\]
\[
x = 9 \text{ 或 } x = -2
\]
\[
\text{因為} 7^y \text{總是正數，所以} x \neq -2
\]
\[
\text{所以} x = 9 \text{即} 7^y = 9 \text{，將其轉為對數形式得：}
\]
\[
y = \log_7 9
\]
\[
\text{所以答案是 C. } \log_7 9
\]
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