4^x = 5^x+1    solve using log ..and tell me -7.25 is the correct answer ?

To solve the equation \( 4^x = 5^{x+1} \) using logarithms, we'll follow these steps:

1. **Take the logarithm of both sides:**  
   \( \log(4^x) = \log(5^{x+1}) \)

2. **Use the logarithm power rule (\(\log(a^b) = b\log(a)\)):**  
   \( x\log(4) = (x+1)\log(5) \)

3. **Expand the right side:**  
   \( x\log(4) = x\log(5) + \log(5) \)

4. **Rearrange the equation to isolate \(x\):**  
   \( x\log(4) - x\log(5) = \log(5) \)

5. **Factor out \(x\):**  
   \( x(\log(4) - \log(5)) = \log(5) \)

6. **Solve for \(x\):**  
   \( x = \frac{\log(5)}{\log(4) - \log(5)} \)

Now, let's calculate the value:

\[
x = \frac{\log(5)}{\log(4) - \log(5)}
\]

Let's compute this value.

First, calculate \(\log(5)\), \(\log(4)\), and then \(x\).The calculated value of \( x \) is approximately \(-7.213\).

Thus, \(-7.25\) is a close approximation, but the exact answer is closer to \(-7.213\).

Would you like further details on the solution or have any other questions?

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### Additional Questions
1. How would you solve \( 3^x = 2^{x+2} \) using logarithms?
2. Can you find the value of \( x \) in the equation \( 2^x = 3^{2x-1} \) using logarithms?
3. What is the solution for \( 6^x = 7^{x+1} \) using logarithms?
4. How do you solve \( 10^x = 2^{2x} \) using logarithms?
5. Can you find \( x \) in \( 2^{3x} = 8^{x+2} \) using logarithms?
6. What is the value of \( x \) in the equation \( 5^{2x} = 3^{x+3} \)?
7. How would you solve \( 2^{x+1} = 5 \) using logarithms?
8. Can you find \( x \) in \( 4^{x-2} = 9 \) using logarithms?

### Tip
When solving exponential equations using logarithms, always consider taking the logarithm of both sides to bring the exponent down and facilitate solving for the variable.

Learn how to solve the exponential equation 4^x = 5^(x+1) using logarithms with a step-by-step solution. Understand the application of logarithm power rules and solve for the unknown variable x. Suitable for high school students.

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