https://file.aiquickdraw.com/mathbot/file/1727722677113avzsncai.jpg

The expression in the image is:

\[
d \cdot \ln(e^4) - \lg(100)
\]

Let's break it down step by step:

### Step 1: Simplify \(\ln(e^4)\)
The natural logarithm of \(e^x\) is a fundamental property of logarithms, where:
\[
\ln(e^x) = x
\]
So, \(\ln(e^4)\) simplifies to:
\[
\ln(e^4) = 4
\]

### Step 2: Simplify \(\lg(100)\)
The \(\lg\) function represents the common logarithm, or logarithm base 10. We know that:
\[
\lg(100) = \log_{10}(100)
\]
Since \(100 = 10^2\), we can simplify:
\[
\log_{10}(100) = 2
\]

### Step 3: Substitute the simplified values into the expression
Now, substitute the simplified results back into the original expression:
\[
d \cdot 4 - 2
\]
This simplifies further to:
\[
4d - 2
\]

### Final Answer:
The simplified expression is:
\[
4d - 2
\]

Let me know if you'd like more details or if you have any questions!

---

Here are five related questions to help you explore further:

1. How would the expression change if the power of \(e\) were different (e.g., \(\ln(e^5)\))?
2. What is the value of the expression if \(d = 1\)?
3. How does the \(\lg(x)\) function differ from the \(\ln(x)\) function in terms of logarithmic bases?
4. How would you differentiate or integrate this expression with respect to \(d\)?
5. What would happen if you replaced the common logarithm with the natural logarithm in \(\lg(100)\)?

**Tip:** When simplifying logarithmic expressions, always look for properties of exponents and logarithms that can make the process faster!

This problem explores how to simplify the expression d * ln(e^4) - lg(100) using logarithmic properties. Follow step-by-step to break down ln(e^4) and lg(100) for a final result of 4d - 2. Suitable for high school students in Grades 9-12.

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