https://file.aiquickdraw.com/mathbot/file/1726169980023vxer08v2.jpg

To expand the logarithm \(\log 6x^3\) using the properties of logarithms and express the final answer in terms of \(\log x\), we will apply the following properties:

1. **Product Rule:** \(\log(ab) = \log a + \log b\)
2. **Power Rule:** \(\log(a^b) = b \cdot \log a\)

Let's expand the expression step-by-step:

1. Start with the given expression:
   \[
   \log 6x^3
   \]

2. Apply the **Product Rule** to separate the logarithm:
   \[
   \log 6x^3 = \log 6 + \log x^3
   \]

3. Apply the **Power Rule** to the term \(\log x^3\):
   \[
   \log x^3 = 3 \cdot \log x
   \]

   Substituting this back into the expression gives:
   \[
   \log 6x^3 = \log 6 + 3 \cdot \log x
   \]

4. Since the problem asks for the answer in terms of \(\log x\), we can express the answer as:
   \[
   \log 6x^3 = \log 6 + 3 \log x
   \]

So, the fully expanded expression in terms of \(\log x\) is:
\[
\boxed{\log 6 + 3 \log x}
\]

If you have any questions or want more details, feel free to ask!

### Related Questions:
1. How do you apply the properties of logarithms to simplify expressions?
2. What are some common applications of logarithms in real-world scenarios?
3. How does the base of a logarithm affect its properties?
4. What is the change of base formula for logarithms, and how is it used?
5. How can logarithmic equations be solved for a specific variable?

**Tip:** When expanding or condensing logarithms, always look for products, quotients, or powers that can be separated using the properties of logarithms.

Expand the logarithm fully using the properties of logs. Express the final answer in terms of log x. log 6x^3

Learn how to fully expand the logarithmic expression log 6x^3 using the product and power rules of logarithms. This problem explains the key properties of logarithms and provides a detailed solution, expressing the final answer in terms of log x. Ideal for students in Grades 9-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation