https://file.aiquickdraw.com/mathbot/file/1731629557993k7sougi9.jpg

Let's work through the problem step-by-step.

We have:
- \( u = \ln a \)
- \( v = \ln b \)

We need to express \( \ln \left( \frac{a^{4/3}}{b} \right) \) in terms of \( u \) and \( v \) without using the \( \ln \) function explicitly.

### Step 1: Expand the Logarithmic Expression
Using logarithmic properties, we can expand:
\[
\ln \left( \frac{a^{4/3}}{b} \right) = \ln \left( a^{4/3} \right) - \ln(b)
\]

### Step 2: Simplify Each Term
Now apply the power rule of logarithms, \( \ln(x^k) = k \ln(x) \):
\[
\ln \left( a^{4/3} \right) = \frac{4}{3} \ln(a)
\]

Since \( \ln(a) = u \) and \( \ln(b) = v \), we substitute:
\[
\ln \left( \frac{a^{4/3}}{b} \right) = \frac{4}{3} u - v
\]

### Final Answer
The expression in terms of \( u \) and \( v \) is:
\[
\ln \left( \frac{a^{4/3}}{b} \right) = \frac{4}{3} u - v
\]

Would you like further details on any of these steps, or do you have any questions?

### Related Questions
1. How do we apply the properties of logarithms in more complex expressions?
2. Can the properties of exponents be used in other logarithmic transformations?
3. What is the significance of the power rule in logarithmic simplification?
4. How would we handle this expression if given different bases for \( a \) and \( b \)?
5. Are there any shortcuts for recognizing when to use \( \ln \) properties in expressions like these?

### Tip
Remember, the power rule for logarithms \( \ln(x^k) = k \ln(x) \) is especially useful for simplifying expressions where the argument of \( \ln \) includes exponents.

Let u = ln a and v = ln b. Write the expression ln( (a^(4/3)) / b ) in terms of u and v without using the ln function.

Learn how to express ln((a^(4/3)) / b) using the variables u and v, where u = ln a and v = ln b. This solution involves the application of logarithmic properties and algebraic simplification, suitable for high school students in grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation