Simplify log(40.5) base 3 - log(10.5) base 3

Step 1: Apply the logarithm property $\log_b(x) - \log_b(y) = \log_b\left(\frac{x}{y}\right)$. The given expression is $\log_3(40.5) - \log_3(10.5)$. $\log_3(40.5) - \log_3(10.5) = \log_3\left(\frac{40.5}{10.5}\right)$

Step 2: Simplify the fraction inside the logarithm. To simplify $\frac{40.5}{10.5}$, multiply the numerator and denominator by 10 to remove the decimals. $\frac{40.5}{10.5} = \frac{40.5 \times 10}{10.5 \times 10} = \frac{405}{105}$ Divide both the numerator and the denominator by their greatest common divisor. Both are divisible by 5. $\frac{405 \div 5}{105 \div 5} = \frac{81}{21}$ Both 81 and 21 are divisible by 3. $\frac{81 \div 3}{21 \div 3} = \frac{27}{7}$

Step 3: Substitute the simplified fraction back into the logarithm. $\log_3\left(\frac{27}{7}\right)$

Step 4: Apply the logarithm property $\log_b\left(\frac{x}{y}\right) = \log_b(x) - \log_b(y)$. $\log_3\left(\frac{27}{7}\right) = \log_3(27) - \log_3(7)$

Step 5: Evaluate $\log_3(27)$. Since $27 = 3^3$, we have $\log_3(27) = 3$.

Step 6: Substitute the value of $\log_3(27)$ into the expression. $3 - \log_3(7)$ Assuming the notation $\log_3^7$ in the options means $\log_3(7)$, the simplified expression matches option C.

The final answer is $\boxed{3 - \log_3^7}$.