We are tasked with finding $\log_{10} \left( \frac{49}{4} \right)$ without using a calculator, given that:

$\log_{10} 7 = 0.8451 \quad \text{and} \quad \log_{10} 2 = 0.3010$

Steps to solve:

1. Use the quotient rule of logarithms: $\log_{10} \left( \frac{49}{4} \right) = \log_{10} 49 - \log_{10} 4$

2. Break down $49$ and $4$ using properties of logarithms: $\log_{10} 49 = \log_{10} \left( 7^2 \right) = 2 \log_{10} 7$ $\log_{10} 4 = \log_{10} \left( 2^2 \right) = 2 \log_{10} 2$

3. Substitute the given values: $\log_{10} 49 = 2 \times 0.8451 = 1.6902$ $\log_{10} 4 = 2 \times 0.3010 = 0.6020$

4. Now calculate $\log_{10} \left( \frac{49}{4} \right)$: $\log_{10} \left( \frac{49}{4} \right) = 1.6902 - 0.6020 = 1.0882$

Thus, the value of $\log_{10} \left( \frac{49}{4} \right)$ is 1.0882.

Would you like more details or have any questions? Here are some related questions to deepen your understanding:

1. How can you use logarithmic properties to simplify other expressions involving fractions?
2. What is the general quotient rule for logarithms, and why does it work?
3. How would you handle a similar problem if the numbers inside the logarithm were more complex (e.g., higher powers)?
4. Can you explain how the change of base formula could be used here if you were not given $\log_{10} 7$?
5. How could you estimate the value of logarithms for numbers between those you are given?

Tip: Always try to break down large numbers into smaller factors or powers to make logarithmic calculations easier.