To solve the equation $2^x = 5^{x+1}$, we can follow these steps:

Step 1: Rewrite the equation

We start by rewriting the equation as: $2^x = 5^{x+1} = 5^x \cdot 5^1 = 5^x \cdot 5$

Step 2: Divide both sides by $5^x$

Now, divide both sides of the equation by $5^x$: $\frac{2^x}{5^x} = 5$ $\left( \frac{2}{5} \right)^x = 5$

Step 3: Take the natural logarithm of both sides

Take the natural logarithm (ln) on both sides: $\ln\left( \left( \frac{2}{5} \right)^x \right) = \ln(5)$ By using the logarithm power rule $\ln(a^b) = b \ln(a)$, this simplifies to: $x \ln\left( \frac{2}{5} \right) = \ln(5)$

Step 4: Solve for $x$

Now, isolate $x$: $x = \frac{\ln(5)}{\ln\left( \frac{5}{2} \right)}$

Step 5: Approximate the value

Using a calculator to approximate the logarithms: $x \approx \frac{\ln(5)}{\ln\left( \frac{5}{2} \right)} \approx \frac{1.6094}{0.9163} \approx 1.756$

Thus, the approximate solution is: $x \approx 1.756$

Would you like further details on any step, or have additional questions?

Here are 5 related questions for further exploration:

1. How can we solve the equation $2^x = 7^{x+2}$?
2. What happens when we change the base to logarithms in equations like $a^x = b^{x+c}$?
3. Can logarithms be used to solve exponential equations with complex numbers?
4. How does the value of $x$ change if we modify the equation to $2^x = 5^x + 1$?
5. What is the graphical interpretation of solving $2^x = 5^{x+1}$?

Tip: Always take the natural logarithm (ln) or logarithm of both sides when you have an exponential equation to simplify it.