https://file.aiquickdraw.com/mathbot/file/1726584211053ykb4shti.jpg

The equation shown in the image is:

\[
\log{x} 2 + \log{x} 3 = 5
\]

I assume the expression involves logarithms with base \(x\), which can be written as:

\[
\log_x 2 + \log_x 3 = 5
\]

To solve this, let's use the logarithmic property:

\[
\log_x a + \log_x b = \log_x (a \cdot b)
\]

Applying this property, we combine the two logarithms:

\[
\log_x (2 \cdot 3) = 5
\]

Simplifying:

\[
\log_x 6 = 5
\]

Now, we can rewrite this logarithmic equation in its exponential form:

\[
x^5 = 6
\]

Finally, to solve for \(x\), we take the fifth root of both sides:

\[
x = \sqrt[5]{6}
\]

Thus, the solution is:

\[
x = 6^{1/5}
\]

Let me know if you need further clarification or a more detailed solution!

Here are 5 related questions to explore:

1. What are the properties of logarithms and how do they help in solving equations?
2. How do you convert a logarithmic equation to its exponential form?
3. Can this type of equation have more than one solution?
4. What happens if we change the base of the logarithms? How does it affect the solution?
5. How do we approximate irrational roots like \( \sqrt[5]{6} \)?

**Tip:** When solving logarithmic equations, always consider the domain restrictions, as the argument of a logarithm must be positive.

Learn how to solve the logarithmic equation log_x 2 + log_x 3 = 5 with detailed step-by-step explanations. This problem involves logarithmic identities and converting logarithms to exponential form. Suitable for high school students in Grades 10-12.

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