To solve the equation \( 100 \cdot 3^x = 2^x \), follow these steps:

1. **Isolate the exponentials**:
   \[
   100 \cdot 3^x = 2^x
   \]
   Divide both sides by \( 3^x \):
   \[
   100 = \left(\frac{2}{3}\right)^x
   \]

2. **Take the natural logarithm (or logarithm of any base) on both sides**:
   \[
   \ln(100) = \ln\left(\left(\frac{2}{3}\right)^x\right)
   \]

3. **Use the power rule for logarithms**:
   \[
   \ln(100) = x \ln\left(\frac{2}{3}\right)
   \]

4. **Solve for \( x \)**:
   \[
   x = \frac{\ln(100)}{\ln\left(\frac{2}{3}\right)}
   \]

Since \(\ln\left(\frac{2}{3}\right)\) is a negative number, \( x \) will be negative.

Let's calculate this using the natural logarithm:

\[
x = \frac{\ln(100)}{\ln\left(\frac{2}{3}\right)}
\]

Now, compute the values:

\[
\ln(100) = \ln(10^2) = 2 \ln(10) \approx 2 \times 2.302585 = 4.60517
\]

\[
\ln\left(\frac{2}{3}\right) = \ln(2) - \ln(3) \approx 0.693147 - 1.098612 = -0.405465
\]

Therefore:

\[
x = \frac{4.60517}{-0.405465} \approx -11.363
\]

So the solution to the equation \( 100 \cdot 3^x = 2^x \) is approximately:

\[
x \approx -11.363
\]

### Questions to Consider:
1. What are the properties of logarithms that are useful for solving exponential equations?
2. How can you check if your solution to an exponential equation is correct?
3. What is the relationship between exponential and logarithmic functions?
4. How do you handle equations with exponential terms on both sides but different bases?
5. Can you solve exponential equations using other bases for logarithms (e.g., common logarithms)?
6. How can you transform an exponential equation into a linear form?
7. What role does the natural logarithm play in solving exponential growth and decay problems?
8. How does changing the base of an exponential function affect the solution to an equation?

**Tip:** When dealing with exponential equations, taking the logarithm of both sides is a powerful technique to simplify and solve for the variable.

Learn how to solve the equation 100 * 3^x = 2^x using logarithms and exponential growth concepts. This problem explores exponential equations and their solutions involving different bases. Suitable for high school students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation