Solve for x and y.{log𝑥+log𝑦2=3log𝑥3+log𝑦3=6

To solve for \(x\) and \(y\) given the system of logarithmic equations:

\[
\begin{cases}
\log x + \log y^2 = 3 \\
\log x^3 + \log y^3 = 6
\end{cases}
\]

we can follow these steps:

1. Use logarithm properties to simplify the equations:
   - \(\log x + \log y^2\) can be rewritten as \(\log(x) + 2\log(y)\)
   - \(\log x^3 + \log y^3\) can be rewritten as \(3\log(x) + 3\log(y)\)

   So the system becomes:
   \[
   \begin{cases}
   \log(x) + 2\log(y) = 3 \\
   3\log(x) + 3\log(y) = 6
   \end{cases}
   \]

2. Simplify the second equation by dividing all terms by 3:
   \[
   \log(x) + \log(y) = 2
   \]

3. Now we have the system:
   \[
   \begin{cases}
   \log(x) + 2\log(y) = 3 \\
   \log(x) + \log(y) = 2
   \end{cases}
   \]

4. Subtract the second equation from the first equation to eliminate \(\log(x)\):
   \[
   (\log(x) + 2\log(y)) - (\log(x) + \log(y)) = 3 - 2
   \]
   \[
   \log(y) = 1
   \]

5. Solve for \(y\):
   \[
   y = 10^1 = 10
   \]

6. Substitute \(\log(y) = 1\) back into the second equation:
   \[
   \log(x) + 1 = 2
   \]
   \[
   \log(x) = 1
   \]

7. Solve for \(x\):
   \[
   x = 10^1 = 10
   \]

Thus, the solutions to the system are:
\[
x = 10 \quad \text{and} \quad y = 10
\]

Do you want details or have any questions?

Here are some more questions you might want to ask:
1. What are the properties of logarithms?
2. How do you solve equations involving logarithms?
3. Can you explain how to change the base of a logarithm?
4. What are some real-life applications of logarithmic functions?
5. How do you graph logarithmic functions?

**Tip:** When working with logarithmic equations, using properties of logarithms such as product, quotient, and power rules can help simplify the equations and make solving easier.

Learn how to solve logarithmic equations log(x) + log(y)^2 = 3 and log(x)^3 + log(y)^3 = 6 using logarithm properties. This problem covers fundamental concepts of logarithms and equations, suitable for high school students.

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