To solve the equation \(1.0375^x = 2\), we can use logarithms. Here's the step-by-step process:

### Step 1: Take the logarithm of both sides
Take the logarithm (common logarithm or natural logarithm) of both sides of the equation:

\[
\log(1.0375^x) = \log(2)
\]

### Step 2: Use the logarithm power rule
The logarithm power rule states that \(\log(a^b) = b \cdot \log(a)\). Apply this rule:

\[
x \cdot \log(1.0375) = \log(2)
\]

### Step 3: Solve for \(x\)
Now, isolate \(x\) by dividing both sides of the equation by \(\log(1.0375)\):

\[
x = \frac{\log(2)}{\log(1.0375)}
\]

### Step 4: Calculate the value of \(x\)
Using a calculator:

\[
x = \frac{\log(2)}{\log(1.0375)} \approx \frac{0.3010}{0.0161} \approx 18.693
\]

So, the solution is:

\[
x \approx 18.693
\]

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

Here are 5 related questions:
1. How would you solve \(2^x = 10\)?
2. What is the general method for solving exponential equations of the form \(a^x = b\)?
3. How does changing the base in an exponential equation affect the solution?
4. Can logarithms be used to solve equations with a base of \(e\) (the natural logarithm)?
5. What is the relationship between logarithms and exponents?

**Tip:** Remember that logarithms are the inverse operations of exponentiation. This property is very useful in solving equations where the unknown variable is in the exponent.

Learn how to solve the exponential equation 1.0375^x = 2 using logarithms. This problem covers exponential functions and logarithms, specifically the logarithm power rule. Suitable for advanced high school students.

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