https://file.aiquickdraw.com/mathbot/file/1727432553738r603ltx9.jpg

The equation in the image appears to be:

\[
e \cdot \log_x{16} = \frac{2}{3}
\]

To solve for \(x\), we need to isolate \(x\) in the logarithmic term. Here's how to proceed step-by-step:

### Step 1: Divide by \(e\)
First, divide both sides of the equation by \(e\):

\[
\log_x{16} = \frac{2}{3e}
\]

### Step 2: Convert logarithm to exponential form
The logarithmic expression \(\log_x{16} = \frac{2}{3e}\) can be rewritten in exponential form:

\[
x^{\frac{2}{3e}} = 16
\]

### Step 3: Solve for \(x\)
To isolate \(x\), raise both sides to the reciprocal of \(\frac{2}{3e}\), which is \(\frac{3e}{2}\):

\[
x = 16^{\frac{3e}{2}}
\]

Thus, \(x = 16^{\frac{3e}{2}}\).

Would you like further details on solving logarithmic or exponential equations?

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Here are 5 related questions to deepen your understanding:
1. What are the properties of logarithms that help simplify expressions like \(\log_x a\)?
2. How does changing the base of a logarithm affect its value?
3. What is the significance of Euler’s number \(e\) in natural logarithms?
4. How can you estimate the value of \(16^{\frac{3e}{2}}\) numerically?
5. How do logarithmic and exponential functions relate in terms of inverses?

**Tip**: When solving logarithmic equations, always remember that logarithms and exponents are inverse operations. Converting between the two is a powerful tool.

Learn how to solve the equation e log_x 16 = 2/3 step by step. This problem involves key logarithmic and exponential concepts, including rewriting logarithmic expressions and using the power rule for exponents. Suitable for students in Grades 10-12.

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