Question

question
find the numerical value of the log expression.
\log a = -3 \quad \log b = -9 \quad \log c = 3
\log \frac{a^7 b^6}{c^2}

Explanation:

Step1: Apply Logarithm Properties

Use the properties of logarithms: $\log\frac{M}{N}=\log M - \log N$ and $\log(MN)=\log M+\log N$, and $\log M^k = k\log M$. So, $\log\frac{a^{7}b^{6}}{c^{2}}=\log(a^{7}b^{6})-\log(c^{2}) = 7\log a + 6\log b-2\log c$.

Step2: Substitute Given Values

We know $\log a = - 3$, $\log b=-9$, $\log c = 3$. Substitute these values into the expression: $7\times(-3)+6\times(-9)-2\times3$.

Step3: Calculate the Result

First, calculate each term: $7\times(-3)=-21$, $6\times(-9)=-54$, $2\times3 = 6$. Then sum them up: $-21-54 - 6=-81$.

Answer:

-81