Problem
Evaluate exactly the expression given that $\log M=2.7$ and $\log N=-3.6$. \[ \log \left(0.1 M^{5} N^{4}\right)= \] eTextbook and Media Hint Save for Later
Solution
Step 1 ：We are given that $\log M = 2.7$ and $\log N = -3.6$.
Step 2 ：We are asked to evaluate the expression $\log \left(0.1 M^{5} N^{4}\right)$.
Step 3 ：We know that $\log (ab) = \log a + \log b$ and $\log a^n = n \log a$.
Step 4 ：Applying these properties, we can rewrite the expression as $\log 0.1 + 5 \log M + 4 \log N$.
Step 5 ：Substituting the given values, we get $-1 + 5(2.7) + 4(-3.6)$.
Step 6 ：Evaluating this expression, we get $-1.9$.
Step 7 ：So, the exact value of the expression $\log \left(0.1 M^{5} N^{4}\right)$ is \(\boxed{-1.9}\).
