It appears that you have several prompts that need to be rewritten or solved, but without seeing the specific content or context in the "Captionless Image" sections, I can only assist you with the equation provided. Here’s how to solve the natural logarithm equation:
Solve the Equation:
Given the equation: \[ \ln(3x - 4) = \ln(2x) \]
Since the logarithmic function is one-to-one, we can set the arguments equal to each other: \[ 3x - 4 = 2x \]
Now, isolate \(x\):
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Subtract \(2x\) from both sides: \[ 3x - 2x - 4 = 0 \] Which simplifies to: \[ x - 4 = 0 \]
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Add \(4\) to both sides: \[ x = 4 \]
Check the solution in the original logarithmic expressions:
Substituting \(x = 4\): \[ 3(4) - 4 = 12 - 4 = 8 \] \[ 2(4) = 8 \]
Both sides yield: \(\ln(8) = \ln(8)\), which confirms that the solution is valid.
Therefore, the solution is:
Option 3: 4
If you can provide more details or text from the captionless images for the other prompts, I’ll be happy to help with those too!