If log7.5 = 0.8751, evaluate 2 log75 + log750

1. Analyze the question

We are given that $\log 7.5 = 0.8751$. We need to evaluate the expression $2 \log 75 + \log 750$ using the properties of logarithms.

2. Explain the concept

To solve this, we need to express $75$ and $750$ in terms of $7.5$ using the laws of logarithms:

• Product Rule: $\log(a \times b) = \log a + \log b$
• Power Rule: $n \log a = \log(a^n)$
• Base 10 property: $\log(10) = 1$ and $\log(100) = 2$

Since $75 = 7.5 \times 10$ and $750 = 7.5 \times 100$, we can rewrite the expression.

3. Show the solution

Step 1: Expand the terms using the product rule.

• $\log 75 = \log(7.5 \times 10) = \log 7.5 + \log 10$
• $\log 750 = \log(7.5 \times 100) = \log 7.5 + \log 100$

Step 2: Substitute these into the original expression. The expression is $2 \log 75 + \log 750$. $= 2(\log 7.5 + \log 10) + (\log 7.5 + \log 100)$

Step 3: Simplify using known values. We know $\log 10 = 1$ and $\log 100 = 2$. $= 2(\log 7.5 + 1) + (\log 7.5 + 2)$ $= 2 \log 7.5 + 2 + \log 7.5 + 2$ $= 3 \log 7.5 + 4$

Step 4: Calculate the final value. Substitute $\log 7.5 = 0.8751$: $= 3(0.8751) + 4$ $= 2.6253 + 4$ $= 6.6253$

4. Identify the answer

Comparing our result to the given options, we find that the correct choice is B.

5. Tip

When dealing with log problems involving decimals like $7.5, 75, 750$, always try to break them down into the given value multiplied by powers of $10$ ($10, 100, 1000$). This makes the calculation much faster and avoids the need for a calculator!