walk me through how to simplify $$ \log_2\left(y\right)=3\log_2\left(x\right)-2 $$

To simplify the equation $\log_2\left(y\right) = 3\log_2\left(x\right) - 2$, we can follow these steps:

### Step-by-Step Solution:

1. **Rewrite the equation using properties of logarithms:**

   The equation given is:
   \[
   \log_2(y) = 3\log_2(x) - 2.
   \]

2. **Apply the power rule of logarithms:**

   Recall that the power rule for logarithms states:
   \[
   a \log_b(c) = \log_b(c^a).
   \]
   Using this rule, rewrite $3\log_2(x)$ as $\log_2(x^3)$:
   \[
   \log_2(y) = \log_2(x^3) - 2.
   \]

3. **Convert the subtraction into division using the logarithm rule:**

   Recall that for logarithms, the subtraction rule is:
   \[
   \log_b(a) - \log_b(c) = \log_b\left(\frac{a}{c}\right).
   \]
   We can rewrite the equation using this property. The number $2$ can be written as a logarithm:
   \[
   \log_2(2^2) = 2.
   \]
   So, rewrite the equation as:
   \[
   \log_2(y) = \log_2(x^3) - \log_2(4).
   \]
   Now, combine the logs:
   \[
   \log_2(y) = \log_2\left(\frac{x^3}{4}\right).
   \]

4. **Exponentiate both sides to remove the logarithm:**

   Since the bases of the logarithms are the same, we can equate the arguments:
   \[
   y = \frac{x^3}{4}.
   \]

### Simplified Result:

The simplified expression is:
\[
y = \frac{x^3}{4}.
\]

This is the expression in its simplest form. The original logarithmic equation is now rewritten as an algebraic expression.

Would you like more details on any step or do you have any questions?  

Here are 5 related questions for further exploration:

1. How do you solve a logarithmic equation that involves different bases?
2. How can you change the base of a logarithm?
3. What is the logarithmic form of an exponential equation?
4. How can you solve logarithmic inequalities?
5. What are the applications of logarithmic functions in real life?

**Tip:** Always remember that logarithmic rules like the power rule, product rule, and quotient rule are powerful tools for simplifying complex logarithmic expressions!

Learn how to simplify the logarithmic equation $\log_2(y) = 3\log_2(x) - 2$ with a step-by-step solution. This problem involves logarithmic properties and the power rule of logarithms. Suitable for high school students studying advanced algebra.

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