This solution explains how to find the value of $log 36$ using the given values for $log 2$ and $log 3$. We will use the fundamental properties of logarithms to break down the problem and arrive at the correct answer.

Logarithm Properties Explained

To solve this problem, we need to understand two key properties of logarithms:

• Product Rule: The logarithm of a product of two numbers is the sum of their logarithms. Mathematically, this is written as: $log(a \times b) = log a + log b$
• Power Rule: The logarithm of a number raised to an exponent is the exponent times the logarithm of the number. Mathematically, this is written as: $log(a^n) = n \times log a$

We are given: $log 2 = 0.3010$ $log 3 = 0.4771$

We need to find the value of $log 36$.

Calculating Log 36 Step-by-Step

First, let's express the number 36 in terms of its prime factors. The prime factors of 36 are 2 and 3.

We can write 36 as:

$36 = 6 \times 6$

Since $6 = 2 \times 3$, we can rewrite 36 as:

$36 = (2 \times 3) \times (2 \times 3)$

Using exponents, we can simplify this expression:

$36 = 2^2 \times 3^2$

Derivation for Log 36

Now, we can apply the logarithm properties to $log 36$.

1. Substitute the prime factorization: $log 36 = log (2^2 \times 3^2)$
2. Apply the Product Rule: Since $log(a \times b) = log a + log b$, we can separate the terms: $log 36 = log (2^2) + log (3^2)$
3. Apply the Power Rule: Since $log(a^n) = n \times log a$, we can bring the exponents down: $log 36 = (2 \times log 2) + (2 \times log 3)$
4. Substitute the given values: Now, we substitute the provided values for $log 2$ and $log 3$: $log 36 = (2 \times 0.3010) + (2 \times 0.4771)$
5. Perform the multiplication: $2 \times 0.3010 = 0.6020$ $2 \times 0.4771 = 0.9542$
6. Perform the addition: $log 36 = 0.6020 + 0.9542$ $log 36 = 1.5562$

Therefore, the value of $log 36$ is $1.5562$. This matches one of the options provided.