The problem you're presenting is a logarithmic expression. From the image, it appears you're given the equation $\log_a b = 4$ and are asked to determine the value of the following expression:

$\log_a \left( \frac{4}{\sqrt{a^b}} \right)$

Let's break this down step by step.

Step 1: Understanding $\log_a b = 4$

This means that the logarithm of $b$ with base $a$ equals 4. We can express this as:

$a^4 = b$

Step 2: Analyzing the expression $\log_a \left( \frac{4}{\sqrt{a^b}} \right)$

We need to evaluate:

$\log_a \left( \frac{4}{\sqrt{a^b}} \right)$

First, let's rewrite $\sqrt{a^b}$ as:

$\sqrt{a^b} = a^{\frac{b}{2}}$

Now, the expression becomes:

$\log_a \left( \frac{4}{a^{\frac{b}{2}}} \right)$

Step 3: Applying Logarithmic Properties

We use the logarithmic property $\log_a \left( \frac{x}{y} \right) = \log_a x - \log_a y$:

$\log_a \left( \frac{4}{a^{\frac{b}{2}}} \right) = \log_a 4 - \log_a \left( a^{\frac{b}{2}} \right)$

We know that $\log_a \left( a^x \right) = x$, so:

$\log_a \left( a^{\frac{b}{2}} \right) = \frac{b}{2}$

Thus, the expression simplifies to:

$\log_a 4 - \frac{b}{2}$

Step 4: Substituting $b = a^4$

From Step 1, we know $b = a^4$, so we substitute this into the expression:

$\log_a 4 - \frac{a^4}{2}$

So, the final result is:

$\log_a 4 - \frac{a^4}{2}$

Would you like to see further details or have any questions about this solution?

Related Questions:

1. How can we use logarithmic identities to simplify more complex expressions?
2. What is the inverse of a logarithmic function, and how can it help in solving equations?
3. How does the logarithm with an exponent work, and how do you simplify such expressions?
4. What is the relationship between logarithms and exponential equations?
5. Can you explain the steps in solving logarithmic equations with different bases?

Tip: When dealing with logarithmic expressions, always try to use basic properties of logarithms such as $\log_b \left( x \cdot y \right) = \log_b x + \log_b y$ and $\log_b \left( \frac{x}{y} \right) = \log_b x - \log_b y$ to break down complex terms into simpler components.