If log$_{10}$2 = 0.3010 and log$_{10}$3 = 0.4771, eventually without using the logarithm tables, log$_{10}$4.5

1. Analyze the question

We are given the values of $\log_{10}2$ and $\log_{10}3$. We need to find the value of $\log_{10}4.5$ by expressing $4.5$ in terms of $2$ and $3$.

2. Explain the concept

To solve this, we use the laws of logarithms:

• Quotient Rule: $\log_b(\frac{x}{y}) = \log_b x - \log_b y$
• Product Rule: $\log_b(xy) = \log_b x + \log_b y$
• Power Rule: $\log_b(x^n) = n \log_b x$

First, convert the decimal $4.5$ into a fraction: $4.5 = \frac{45}{10} = \frac{9}{2}$

3. Show the solution

Step 1: Express $\log_{10}4.5$ using the quotient rule.

Step 2: Express $9$ as a power of $3$. Since $9 = 3^2$, we have:

Step 3: Substitute the known values into the equation.

Step 4: Perform the arithmetic.

4. Identify the answer

Comparing our result to the given options: A. 0.3010 B. 0.4771 C. 0.6532 D. 0.9542

The correct option is C.

5. Tip

Whenever you see a decimal in a logarithm question (like 4.5 or 0.75), always convert it to a fraction first. It makes it much easier to break the number down into the prime factors (2, 3, 5, etc.) provided in the question!