The given problem asks to find $\log 15$ using the provided information:

• $\log 2 = a$
• $\log 3 = b$

Let me solve it step-by-step with detailed explanations:

Step 1: Express $\log 15$ in terms of its prime factors

The number 15 can be expressed as: $15 = 3 \times 5$ Using the logarithmic property: $\log(ab) = \log a + \log b$ we can write: $\log 15 = \log 3 + \log 5$

Step 2: Express $\log 5$ using $\log 2$

Since $\log 2 = a$, we relate $\log 5$ to $\log 2$ by using the property of logarithms for reciprocals: $\log 10 = \log(2 \times 5) = \log 2 + \log 5$ We know that: $\log 10 = 1 \quad (\text{logarithm base 10 of 10 is 1})$ Thus: $1 = \log 2 + \log 5$ Rearranging: $\log 5 = 1 - \log 2$ Substitute $\log 2 = a$: $\log 5 = 1 - a$

Step 3: Substitute values back into $\log 15$

Now substitute $\log 3 = b$ and $\log 5 = 1 - a$ into $\log 15$: $\log 15 = \log 3 + \log 5$ $\log 15 = b + (1 - a)$ Simplify: $\log 15 = b + 1 - a$

Final Answer:

$\log 15 = b + 1 - a$

Key Explanation:

1. The problem uses the basic properties of logarithms:
   • Product property: $\log(ab) = \log a + \log b$
   • Known values like $\log 10 = 1$
2. The key was to express $\log 5$ in terms of $\log 2$ using the fact that $\log 10 = 1$.
3. This ensures that the result depends only on $a$ and $b$, as requested.

Let me know if you'd like me to expand on any of these steps!

Related Questions to Explore:

1. What is the logarithm property for division? How is $\log(a/b)$ calculated?
2. How can we compute $\log(30)$ using $\log 2$, $\log 3$, and $\log 5$?
3. What is the significance of the base in logarithms, and how would this change if the base were not 10?
4. How can we derive $\log(25)$ using $\log 2 = a$ and $\log 5 = 1 - a$?
5. Can we generalize this approach to find the logarithm of any composite number?

Tip:

Always simplify logarithmic expressions step-by-step using the fundamental properties, as it helps avoid mistakes and makes the problem easier to interpret.