If $\log 27 = 1.431$, then the value of $\log 9$ is

### Concept & Formula When dealing with common bases, break down the logarithm arguments into their prime factorization. Both $27$ and $9$ are powers of $3$. By finding the fundamental value of $\log 3$, we can easily compute $\log 9$. The key rule is the Power Rule of Logarithms. Formula: $\log (x^n) = n \log x$ ### Step-by-Step Solution Given the equation: $$ \log 27 = 1.431 $$ Step 1: Rewrite $27$ as a power of $3$. $$ \log (3^3) = 1.431 $$ Step 2: Apply the power rule to bring the exponent down. $$ 3 \log 3 = 1.431 $$ Step 3: Solve for the base value, $\log 3$. $$ \log 3 = \frac{1.431}{3} $$ $$ \log 3 = 0.477 $$ Step 4: Now find the value of $\log 9$. Rewrite $9$ as $3^2$. $$ \log 9 = \log (3^2) $$ $$ \log 9 = 2 \log 3 $$ Step 5: Substitute the value of $\log 3$ we just found. $$ \log 9 = 2 \times 0.477 $$ $$ \log 9 = 0.954 $$ ### Exam Strategy & Shortcut **Direct Ratio Method:** Since both arguments are powers of the same base ($3$), you can set up a direct ratio of their exponents. $\log 27$ involves $3^3$ (exponent $3$) and $\log 9$ involves $3^2$ (exponent $2$). Therefore, the value of $\log 9$ is simply exactly $\frac{2}{3}$ of the value of $\log 27$. $$ \log 9 = \frac{2}{3} \times 1.431 = 2 \times 0.477 = 0.954 $$ ### Common Pitfall A frequent mistake is attempting to guess log values or assuming a linear relationship (e.g., thinking $\log 9$ is one-third of $\log 27$ just because $9$ is one-third of $27$). Logarithmic relationships are strictly exponential, so always convert to powers first! ### Final Answer **Therefore, the correct answer is 0.954.**