To simplify the expression $\log{\frac{1}{16}} \times 4 \log{\sqrt{3}} \times 3 \log{100}$, we'll proceed step by step:

Step 1: Simplify Each Logarithm

• $\log{\frac{1}{16}}$: $\log{\frac{1}{16}} = \log{(16^{-1})} = -\log{16} = -\log{2^4} = -4\log{2}$

• $4 \log{\sqrt{3}}$: $4 \log{\sqrt{3}} = 4 \times \log{3^{1/2}} = 4 \times \frac{1}{2} \log{3} = 2 \log{3}$

• $3 \log{100}$: $3 \log{100} = 3 \log{10^2} = 3 \times 2 \log{10} = 6 \log{10}$ Since $\log{10} = 1$, this simplifies to: $6 \log{10} = 6$

Step 2: Substitute the Simplified Logs Back into the Expression

Substituting the simplified logarithms, the original expression becomes: $(-4 \log{2}) \times (2 \log{3}) \times 6$

Step 3: Combine the Expression

Now, let's combine the expression: $-4 \times 2 \times 6 \times \log{2} \times \log{3}$ $-48 \times \log{2} \times \log{3}$

So, the simplified form of the expression is: $-48 \log{2} \log{3}$

This is the final simplified form of the expression.

Would you like more details on any part of this solution or have any questions?

5 Related Questions:

1. How do you simplify logarithms with fractional exponents?
2. What are the properties of logarithms used in simplification?
3. How do you convert logarithms with different bases?
4. What is the logarithm of a fraction and how is it simplified?
5. Can the product of logarithms be simplified further in some cases?

Tip: Remember, logarithms have properties that can simplify expressions, such as the power rule $\log{a^b} = b \log{a}$.