It looks like the file you uploaded contains logarithmic equations that need to be solved. I'll extract the relevant problem from the document and walk through the solution step-by-step. Give me a moment to work on it.

Let me check for the specific problems now.I've identified the logarithmic equations in your file. Let's go through one example, solving it step-by-step:

Problem:

Solve the equation $\log(x) + \log(x - 1) = 1$.

Step-by-Step Solution:

1. Combine the logs using logarithmic properties:

   By using the property $\log_a M + \log_a N = \log_a(MN)$, we can combine the logarithms: $\log(x) + \log(x - 1) = \log(x(x - 1)) = \log(x^2 - x)$ So the equation becomes: $\log(x^2 - x) = 1$

2. Rewrite the logarithmic equation in exponential form:

   Since $\log_b A = C$ implies $A = b^C$, we rewrite the equation: $x^2 - x = 10^1$ Simplifying: $x^2 - x = 10$

3. Solve the quadratic equation:

   Now we have the quadratic equation: $x^2 - x - 10 = 0$ We can solve this using the quadratic formula: $x = \frac{-(-1) \pm \sqrt{(-1)^2 - 4(1)(-10)}}{2(1)}$ Simplifying: $x = \frac{1 \pm \sqrt{1 + 40}}{2} = \frac{1 \pm \sqrt{41}}{2}$ So: $x = \frac{1 + \sqrt{41}}{2} \quad \text{or} \quad x = \frac{1 - \sqrt{41}}{2}$

4. Check for extraneous solutions:

   Logarithmic equations have restrictions since the argument of a log must be positive. Let's check the two solutions:

   • $x = \frac{1 + \sqrt{41}}{2} \approx 3.70$: This is valid since $x > 1$.
   • $x = \frac{1 - \sqrt{41}}{2} \approx -2.70$: This is not valid because logarithms of negative numbers are undefined.

   Therefore, the solution is: $x = \frac{1 + \sqrt{41}}{2} \approx 3.70$

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Would you like more details or have any questions on this?

Here are 5 related questions:

1. What are the properties of logarithms and how are they used in solving equations?
2. How do you solve quadratic equations by factoring or completing the square?
3. What are extraneous solutions in logarithmic equations, and how do they arise?
4. How do logarithmic and exponential functions relate to each other?
5. Can logarithms have negative numbers as arguments, and why or why not?

Tip: Always check for extraneous solutions when solving logarithmic equations, as not all solutions to the quadratic might satisfy the original equation.