https://file.aiquickdraw.com/mathbot/file/17267745173398gj7ess9.jpg

I can help you solve the equations from the image you uploaded. Let's tackle them one by one. Here is equation **(a)** from the image:

### (a) \( 5^{x-1} = \frac{1}{2} \)

We can solve this equation by applying logarithmic principles or converting the equation into simpler steps. Let's solve it:

\[
5^{x-1} = \frac{1}{2}
\]

Take the natural logarithm (or log base 10) of both sides:

\[
\ln(5^{x-1}) = \ln\left(\frac{1}{2}\right)
\]

Using the logarithmic property \( \ln(a^b) = b\ln(a) \), we get:

\[
(x-1) \ln(5) = \ln\left(\frac{1}{2}\right)
\]

Now, let's compute the value for \( \ln\left(\frac{1}{2}\right) \) and solve for \( x \). I will calculate this next.The solution to equation (a) \( 5^{x-1} = \frac{1}{2} \) is approximately:

\[
x \approx 0.569
\]

Would you like to proceed with solving equation **(b)** next? Also, let me know if you need more details on any of the steps.

Here are some related questions you might find useful:
1. How do you apply logarithms to solve exponential equations?
2. What are the properties of logarithms that simplify complex expressions?
3. How can logarithms be used in different bases (e.g., base 10, base 2)?
4. What is the difference between natural logarithms and common logarithms?
5. Why do we use logarithmic functions in mathematical modeling?

**Tip:** When solving logarithmic or exponential equations, always check for possible restrictions on the variable to ensure valid solutions.

Solve for x in the following equation. Show your complete solution.

Detailed solutions to a variety of exponential and logarithmic equations, including equations involving natural logarithms, change of base, and properties of logarithms. Suitable for students in Grades 10-12.

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