https://r2.erweima.ai/mathbot/image/1726088856114mgnbws5d.jpg

The problem you highlighted is \( \log_a \left( \frac{\sqrt[3]{a}}{a^2} \right) \).

Let's solve this step-by-step:

### Step 1: Simplify the expression inside the logarithm.
The expression inside the logarithm is:

\[
\frac{\sqrt[3]{a}}{a^2}
\]

We can rewrite the cube root of \(a\) as \(a^{1/3}\), so the expression becomes:

\[
\frac{a^{1/3}}{a^2}
\]

### Step 2: Apply the laws of exponents.
Using the exponent rule \(\frac{a^m}{a^n} = a^{m-n}\), we can simplify:

\[
\frac{a^{1/3}}{a^2} = a^{1/3 - 2} = a^{-5/3}
\]

### Step 3: Apply the logarithmic property.
Now we can use the logarithmic identity \(\log_a (a^x) = x\). Therefore:

\[
\log_a \left( a^{-5/3} \right) = -\frac{5}{3}
\]

So, the final answer is:

\[
\boxed{-\frac{5}{3}}
\]

Would you like more details on any step, or do you have any questions?

### Related Questions:
1. What are the properties of logarithms, and how are they used?
2. How do exponent rules apply to different bases?
3. What happens if the base of the logarithm changes?
4. Can you apply logarithmic properties to complex numbers?
5. How do you solve equations involving logarithms with different bases?

#### Tip:
Always simplify the expression inside the logarithm using exponent rules before applying logarithmic properties.

Learn how to simplify the logarithmic expression log_a( (cube root of a) / a^2 ) step by step, using logarithmic identities and the laws of exponents. This problem involves key concepts such as exponent rules and logarithmic properties, suitable for 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