To solve the equation $5 \times \log_5 \left(\frac{1}{125}\right) = x$, let's break it down step by step.

Step 1: Simplify the logarithm

We know that: $\log_b\left(\frac{1}{a}\right) = -\log_b(a)$ Also, $125$ can be written as $5^3$. So: $\log_5\left(\frac{1}{125}\right) = \log_5\left(\frac{1}{5^3}\right)$ Using the property mentioned earlier: $\log_5\left(\frac{1}{5^3}\right) = -\log_5\left(5^3\right)$ And since: $\log_5(5^3) = 3$ We have: $\log_5\left(\frac{1}{125}\right) = -3$

Step 2: Multiply by 5

Now substitute back into the original equation: $5 \times \log_5\left(\frac{1}{125}\right) = 5 \times (-3)$ This simplifies to: $x = -15$

Final Answer

$x = -15$

Would you like more details, or do you have any other questions?

Related Questions:

1. How do logarithm properties simplify equations?
2. What is the general approach to solving logarithmic equations?
3. How does the base of the logarithm affect the solution?
4. Can we generalize the steps to solve $a \times \log_b(c) = x$?
5. What are some real-world applications of logarithms?

Tip: Always remember that logarithms are exponents, which helps in simplifying and solving logarithmic equations.