To solve the equation \( 5^x \cdot 25^x = 100 \), follow these steps:

1. **Express \( 25^x \) in terms of \( 5 \):**

\[ 25 = 5^2 \]
\[ 25^x = (5^2)^x = 5^{2x} \]

So the equation becomes:
\[ 5^x \cdot 5^{2x} = 100 \]

2. **Combine the exponents:**

\[ 5^x \cdot 5^{2x} = 5^{x + 2x} = 5^{3x} \]

So now the equation is:
\[ 5^{3x} = 100 \]

3. **Take the logarithm of both sides:**

\[ \log(5^{3x}) = \log(100) \]

Using the property of logarithms \( \log(a^b) = b\log(a) \), we get:
\[ 3x \log(5) = \log(100) \]

4. **Solve for \( x \):**

\[ x = \frac{\log(100)}{3 \log(5)} \]

5. **Simplify the logarithmic expression:**

We know that \( \log(100) = \log(10^2) = 2 \log(10) = 2 \) (since \( \log(10) = 1 \)).
So,
\[ x = \frac{2}{3 \log(5)} \]

Therefore, the solution to the equation \( 5^x \cdot 25^x = 100 \) is:
\[ x = \frac{2}{3 \log(5)} \]

If you have any questions or would like more details, please let me know!

Here are some related questions for further exploration:
1. How do you solve equations involving exponential functions?
2. What are the properties of logarithms that are useful in solving exponential equations?
3. How do you convert an exponential equation to a logarithmic equation?
4. What is the relationship between exponential and logarithmic functions?
5. How can you solve for an exponent when the base and the result are given?
6. How does changing the base of the logarithm affect the solution of an equation?
7. What are the common bases used in logarithms, and why?
8. How do you interpret the solution of an exponential equation in a real-world context?

**Tip:** When dealing with exponential equations, it often helps to express all terms with the same base, if possible, to simplify the equation.

Learn how to solve the exponential equation 5^x * 25^x = 100 with a step-by-step solution. This problem involves exponential functions and logarithms, emphasizing the properties of exponents and logarithmic transformations. Suitable for high school students.

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