What is the value of ${\log }_7{7}^{{\log }_{7}7}$?

• A. $1$
• B. $7^7$
• C. $7$
• D. ${\log }_{7}7$

Answer: (C)

Verified step by step guidance

1. Identify the expression given: \(\log_{7} \left(7^{\log_{7} 7}\right)\). This involves a logarithm with base 7 and an exponentiation inside the argument.
2. Recall the logarithm power rule: \(\log_{a} (b^{c}) = c \cdot \log_{a} b\). Apply this rule to rewrite the expression as \(\left(\log_{7} 7\right) \cdot \log_{7} 7\).
3. Recognize that \(\log_{7} 7\) is the logarithm of a number to its own base, which equals 1. So, \(\log_{7} 7 = 1\).
4. Substitute this value back into the expression: it becomes \$1 \cdot 1$.
5. Conclude that the value of the original expression is 1, based on the properties of logarithms and exponents.