We are given the following logarithmic values:

• $\log 2 = 0.301$
• $\log 3 = 0.477$
• $\log 5 = 0.699$

Step 1: Express $90$ as a product of its prime factors

$90 = 2 \times 3^2 \times 5$

Using the logarithmic property: $\log(ab) = \log a + \log b \quad \text{and} \quad \log(a^n) = n \log a$

Step 2: Apply the formula

$\log 90 = \log(2 \times 3^2 \times 5)$

Break it down: $\log 90 = \log 2 + \log 3^2 + \log 5$

$\log 90 = \log 2 + 2\log 3 + \log 5$

Step 3: Substitute the given values

$\log 90 = 0.301 + 2(0.477) + 0.699$

Simplify: $\log 90 = 0.301 + 0.954 + 0.699$

$\log 90 = 1.954$

Final Answer:

$\log 90 = 1.954$

Would you like me to elaborate on any step or discuss logarithmic properties further?

Related Questions:

1. What is the logarithm of other numbers like $\log 45$ or $\log 60$ using these values?
2. How are logarithmic properties useful for simplifying calculations without a calculator?
3. Can you demonstrate the use of $\log(ab/c)$ in problem-solving?
4. How can this calculation be used in solving exponential equations?
5. What are the differences between natural logarithms ($\ln$) and common logarithms ($\log$)?

Tip:

Always factorize composite numbers into primes before applying logarithmic properties for easier