https://file.aiquickdraw.com/mathbot/file/1725378482524cepnp04s.jpg

The equation shown in the image is:

\[
\frac{13}{4} = \log_{2x} 2 + \log_{x} x
\]

Let's break this down and solve it step by step.

### Step 1: Simplify the equation
The term \(\log_{x} x\) simplifies directly because \(\log_{b} b = 1\), where \(b\) is the base of the logarithm:

\[
\log_{x} x = 1
\]

So the equation becomes:

\[
\frac{13}{4} = \log_{2x} 2 + 1
\]

### Step 2: Subtract 1 from both sides

\[
\frac{13}{4} - 1 = \log_{2x} 2
\]

Simplify the left side:

\[
\frac{13}{4} - \frac{4}{4} = \frac{9}{4}
\]

So, we now have:

\[
\frac{9}{4} = \log_{2x} 2
\]

### Step 3: Interpret the logarithm
The equation \(\log_{a} b = c\) means that \(a^c = b\). In our case:

\[
(2x)^{\frac{9}{4}} = 2
\]

### Step 4: Solve for \(x\)
To solve this, raise both sides to the power of \(\frac{4}{9}\) to eliminate the exponent on the left side:

\[
2x = 2^{\frac{4}{9}}
\]

Since the bases are the same, the exponents must be equal:

\[
2x = 2^{\frac{4}{9}}
\]

Now, divide both sides by 2:

\[
x = \frac{2^{\frac{4}{9}}}{2} = 2^{\frac{4}{9} - 1} = 2^{-\frac{5}{9}}
\]

So the solution is:

\[
x = 2^{-\frac{5}{9}}
\]

Would you like any further details or explanations?

### Further Questions:
1. How do you simplify logarithmic expressions involving multiplication?
2. What are the properties of logarithms used to solve the equation?
3. How can we interpret the base of a logarithm?
4. What happens if the base of the logarithm is different?
5. How can we express the final result in decimal form?

**Tip:** Understanding logarithm properties, such as the change of base formula, can simplify solving complex logarithmic equations.

Solve the equation: 13/4 = log_(2x) 2 + log_x x

This solution involves simplifying logarithmic expressions and solving the equation 13/4 = log_(2x) 2 + log_x x using key properties of logarithms. Suitable for students in Grades 10-12.

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