https://file.aiquickdraw.com/mathbot/file/1727020463222u8bzibfi.jpg

The expression given in the image is:

\[
\ln \left( \frac{1}{4^k} \right)
\]

We are tasked with expanding this logarithm using properties of logarithms (sum, difference, or product). Let’s break it down:

### Step 1: Use the logarithm of a fraction
The logarithm of a quotient can be written as:

\[
\ln \left( \frac{a}{b} \right) = \ln(a) - \ln(b)
\]

In this case, \( a = 1 \) and \( b = 4^k \), so:

\[
\ln \left( \frac{1}{4^k} \right) = \ln(1) - \ln(4^k)
\]

Since \( \ln(1) = 0 \), we are left with:

\[
0 - \ln(4^k) = - \ln(4^k)
\]

### Step 2: Use the power rule of logarithms
The power rule states:

\[
\ln(a^b) = b \ln(a)
\]

Here, we apply this to \( \ln(4^k) \):

\[
- \ln(4^k) = - k \ln(4)
\]

Thus, the expanded expression is:

\[
\ln \left( \frac{1}{4^k} \right) = -k \ln(4)
\]

This is the fully expanded logarithmic expression.

---

Do you want more details or have any questions?

Here are 5 related questions to deepen your understanding:

1. How would you expand \(\ln \left( \frac{1}{x^2} \right)\)?
2. Can you simplify \(\ln \left( \frac{1}{a^b} \right)\) using the same properties?
3. What is the expanded form of \(\ln(a^b c^d)\)?
4. How does the property \(\ln(ab) = \ln(a) + \ln(b)\) differ from the quotient rule?
5. How does \(\ln(1) = 0\) affect logarithmic expansions involving fractions?

**Tip:** Always look for ways to apply the basic logarithmic properties (product, quotient, and power rules) when simplifying complex logarithmic expressions.

Expand the logarithm as much as possible. Rewrite the expression as a sum, difference, or product of logs: ln(1/4^k).

This solution shows how to expand the logarithmic expression ln(1/4^k) by applying the quotient and power rules of logarithms. The problem involves simplifying the expression step-by-step. Suitable for college students in introductory algebra or precalculus.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation