two times the logarithm of x to the base three plus two times the logarithm of x squared to the base three is equal to 64. Solve this

We are given the equation:

\[
2 \log_3{x} + 2 \log_3{x^2} = 64
\]

### Step 1: Use logarithmic properties
We can simplify the expression using logarithmic identities.

1. **Product Rule**: \(\log_b{(a^m)} = m \log_b{a}\).
2. **Power Rule**: \(\log_b{(a^m)} = m \log_b{a}\).

First, let's simplify each logarithmic term:

\[
2 \log_3{x} + 2 \log_3{x^2}
\]

For the second term, use the Power Rule:

\[
\log_3{x^2} = 2 \log_3{x}
\]

So the equation becomes:

\[
2 \log_3{x} + 2(2 \log_3{x}) = 64
\]

This simplifies to:

\[
2 \log_3{x} + 4 \log_3{x} = 64
\]

\[
6 \log_3{x} = 64
\]

### Step 2: Solve for \(\log_3{x}\)
Now divide both sides by 6:

\[
\log_3{x} = \frac{64}{6} = \frac{32}{3}
\]

### Step 3: Solve for \(x\)
We now rewrite the equation in exponential form:

\[
x = 3^{\frac{32}{3}}
\]

Thus, the solution is:

\[
x = 3^{\frac{32}{3}}
\]

This is the exact solution. If you'd like an approximation, it can be calculated numerically.

Would you like to see the approximate decimal value, more details on logarithmic properties, or have any questions? 

Here are 5 follow-up questions for further understanding:

1. How do you simplify logarithmic expressions using the product and power rules?
2. What is the difference between common logarithms and natural logarithms?
3. How can exponential equations be solved using logarithms?
4. How would you solve if the base of the logarithms was different, say 10 instead of 3?
5. Can you explain the concept of logarithms in terms of inverse functions?

**Tip**: Always remember the key logarithmic identities—they simplify complex expressions and make problem-solving easier!

Two times the logarithm of x to the base three plus two times the logarithm of x squared to the base three is equal to 64. Solve this.

Learn how to solve a logarithmic equation involving base 3, with step-by-step instructions using the power and product rules. This problem includes simplifying logarithmic expressions and converting to exponential form. Suitable for students in Grades 10-12.

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