what is the relationship between a, b, and c if $a=(log_dx)^2$, $b=log_d(x)^2$, $c=log_d(log_dx)$, and $1<x<d$?

Given the definitions of $ a $, $ b $, and $ c $:

\[
a = (\log_d x)^2, \quad b = \log_d(x)^2, \quad c = \log_d(\log_d x)
\]

Let's analyze each expression:

1. **Expression for $ a $:**

   \[
   a = (\log_d x)^2
   \]

   Here, $ a $ is simply the square of $ \log_d x $.

2. **Expression for $ b $:**

   \[
   b = \log_d(x)^2
   \]

   This is slightly different from $ a $. Notice that $ \log_d(x)^2 $ can be written as:

   \[
   b = 2 \log_d x
   \]

   So, $ b $ is **twice** the logarithm of $ x $ to the base $ d $.

3. **Expression for $ c $:**

   \[
   c = \log_d(\log_d x)
   \]

   Here, $ c $ is the logarithm of $ \log_d x $ to the base $ d $.

### Relationship Between $ a $, $ b $, and $ c $:

We now have:

\[
a = (\log_d x)^2, \quad b = 2 \log_d x, \quad c = \log_d(\log_d x)
\]

Let's express everything in terms of $ \log_d x $, which we will denote as $ L $:

\[
a = L^2, \quad b = 2L, \quad c = \log_d L
\]

The relationships can be summarized as follows:

1. **Relationship between $ a $ and $ b $:**
   \[
   a = \frac{b^2}{4}
   \]
   since $ b = 2L $ and $ a = L^2 $.

2. **Relationship between $ c $ and $ b $:**
   \[
   c = \log_d \left(\frac{b}{2}\right)
   \]
   because $ L = \frac{b}{2} $ and $ c = \log_d L $.

3. **Relationship between $ c $ and $ a $:**
   \[
   c = \log_d \sqrt{a} = \frac{1}{2} \log_d a
   \]

### Summary:

- $ a = (\log_d x)^2 $ (quadratic relationship with $ \log_d x $).
- $ b = 2 \log_d x $ (linear relationship with $ \log_d x $).
- $ c = \log_d(\log_d x) $ (logarithmic relationship with $ \log_d x $).

These expressions establish how $ a $, $ b $, and $ c $ are related in terms of logarithmic and quadratic functions. Specifically:

\[
a = \frac{b^2}{4}, \quad c = \log_d \left(\frac{b}{2}\right), \quad c = \frac{1}{2} \log_d a
\]

Would you like more details or have further questions?

### Related Questions:

1. How would the relationships change if $ 0 < x < 1 $?
2. What is the relationship between $ a $, $ b $, and $ c $ if $ d > x > 1 $?
3. How do these relationships change if $ d = e $ (natural logarithm)?
4. What happens if $ d < 1 $?
5. Can we express $ c $ directly in terms of $ a $ and $ b $?

### Tip:
When dealing with logarithmic expressions, be careful about the placement of exponents—small differences can lead to significantly different interpretations.

Explore the intricate relationships between logarithms and quadratic functions with detailed explanations of variables $a$, $b$, and $c$. Learn how these expressions interact within specified ranges of $1 < x < d$.

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