The value of $\log_{343} 7$ is

Concept & Formula When the base of a logarithm is larger than the argument, expect a fractional exponent. Use the base change or exponent conversion rule: $$ \log_a b = x \implies a^x = b $$ Step-by-Step Solution * **Given:** Evaluate $\log_{343} 7$. * **Calculation:** Let the unknown value be $x$. $$ \log_{343} 7 = x $$ * Convert the logarithmic equation into an exponential equation: $$ 343^x = 7 $$ * Recognize that $343$ is a perfect cube of $7$, meaning $343 = 7^3$. Substitute this into the equation: $$ (7^3)^x = 7^1 $$ $$ 7^{3x} = 7^1 $$ * Since the bases are identical, equate the exponents: $$ 3x = 1 $$ $$ x = \frac{1}{3} $$ Exam Strategy & Shortcut If you memorize cubes up to $10$, you will instantly spot that $7^3 = 343$. Since the logarithm asks "what power of $343$ gives $7$", taking the cube root (power of $1/3$) is the immediate mental answer. Common Pitfall Students often answer with $3$ instead of $1/3$, confusing $\log_{343} 7$ with $\log_7 343$. Always pay attention to which number is the base and which is the argument. Final Answer **Therefore, the correct answer is \frac{1}{3}.**