Suppose log(a) = 9; log(b) = 3 Find log(a ^ 2 * b ^ 4); log(a ^ 2 * b ^ 4) =

To find \(\log(a^2 \cdot b^4)\) given that \(\log(a) = 9\) and \(\log(b) = 3\), we can use the properties of logarithms.

1. **Apply the logarithm to the given expression:**
   \[
   \log(a^2 \cdot b^4)
   \]

2. **Use the property \(\log(x \cdot y) = \log(x) + \log(y)\):**
   \[
   \log(a^2 \cdot b^4) = \log(a^2) + \log(b^4)
   \]

3. **Use the property \(\log(x^y) = y \cdot \log(x)\):**
   \[
   \log(a^2) = 2 \cdot \log(a)
   \]
   \[
   \log(b^4) = 4 \cdot \log(b)
   \]

4. **Substitute the given values \(\log(a) = 9\) and \(\log(b) = 3\):**
   \[
   \log(a^2) = 2 \cdot 9 = 18
   \]
   \[
   \log(b^4) = 4 \cdot 3 = 12
   \]

5. **Add the results:**
   \[
   \log(a^2 \cdot b^4) = 18 + 12 = 30
   \]

Therefore, \(\log(a^2 \cdot b^4) = 30\).

Would you like more details or have any questions?

### Related Questions:
1. How do you use the properties of logarithms to simplify expressions?
2. Can you solve \(\log(x^3 \cdot y^2)\) given \(\log(x) = 4\) and \(\log(y) = 5\)?
3. What is the change of base formula for logarithms?
4. How do you solve equations involving logarithms and exponents?
5. What are some common logarithmic identities used in calculus?
6. Can you explain the relationship between logarithms and exponential functions?
7. How would you solve \(\log(x) + \log(2x - 3) = 5\)?
8. What is the logarithm of a product, quotient, or power?

### Tip:
When working with logarithms, always remember to use the fundamental properties of logarithms such as the product rule, quotient rule, and power rule to simplify and solve expressions efficiently.

Learn how to find \(\log(a^2 \cdot b^4)\) using the properties of logarithms. Given \(\log(a) = 9\) and \(\log(b) = 3\), apply the logarithm to the given expression and use the logarithm properties to simplify step-by-step. Suitable for high school students learning logarithms.

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