https://file.aiquickdraw.com/mathbot/file/1728932719941rt9tgk9e.jpg

The equation in the image appears to be:

\[
5^x = 2^{x+1}
\]

Let's solve this equation step by step.

### Step 1: Express both sides in logarithmic form
To solve for \(x\), we can take the logarithm of both sides of the equation. Using natural logarithms (ln):

\[
\ln(5^x) = \ln(2^{x+1})
\]

### Step 2: Apply the logarithmic power rule
The logarithmic power rule states that \(\ln(a^b) = b \ln(a)\). Applying this rule to both sides:

\[
x \ln(5) = (x+1) \ln(2)
\]

### Step 3: Expand and simplify
Distribute the \(\ln(2)\) on the right-hand side:

\[
x \ln(5) = x \ln(2) + \ln(2)
\]

### Step 4: Isolate the terms with \(x\)
Move the \(x \ln(2)\) term to the left-hand side to group the \(x\) terms together:

\[
x \ln(5) - x \ln(2) = \ln(2)
\]

Factor out \(x\) on the left-hand side:

\[
x (\ln(5) - \ln(2)) = \ln(2)
\]

### Step 5: Solve for \(x\)
Finally, solve for \(x\) by dividing both sides by \((\ln(5) - \ln(2))\):

\[
x = \frac{\ln(2)}{\ln(5) - \ln(2)}
\]

### Step 6: Numerical solution
We can now compute the numerical value of \(x\) using the logarithms:

\[
x = \frac{\ln(2)}{\ln(5) - \ln(2)} \approx \frac{0.6931}{1.6094 - 0.6931} \approx \frac{0.6931}{0.9163} \approx 0.7562
\]

Thus, the solution is approximately:

\[
x \approx 0.756
\]

Would you like further details or have any questions about this solution?

Here are five related questions for deeper understanding:
1. How does changing the base of logarithms affect the solution?
2. Can this equation be solved using exponential properties instead of logarithms?
3. How can this process be applied to equations with different bases?
4. What if the equation was \(5^x = 3^{x+1}\)? How would the solution change?
5. How would you verify the solution using a calculator?

**Tip:** When solving exponential equations, logarithms are a powerful tool to simplify and isolate the variable.

This problem explains how to solve the exponential equation 5^x = 2^{x+1} using logarithms and the logarithmic power rule. Learn step-by-step how to isolate x and solve numerically. Suitable for students in Grades 10-12.

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