Final answer:
The equation x - 1/(x² + 4) = ln(x - 1) - ln(x² + 4) is solved by recognizing logarithmic properties and setting the fraction equal to its natural exponential equivalent. This results in x = 1 + 1/(x² + 4), which can be solved further by clearing the denominator and finding x's values.
Step-by-step explanation:
To solve the equation x - 1/(x² + 4) = ln(x - 1) - ln(x² + 4) for x, we must understand that natural logarithms are the inverse of exponential functions. This knowledge enables us to apply properties of logarithms to simplify the equation. First, recognize that the subtraction of logarithms on the right side of the equation suggests a division within a single logarithm, according to the properties of logarithms.
Combine the logarithms on the right:
ln(x - 1) - ln(x² + 4) = ln((x - 1)/(x² + 4)).
Now the equation becomes x - 1/(x² + 4) = ln((x - 1)/(x² + 4)). At this point, the equation implies that if the natural logarithm of a fraction is equal to the fraction itself, the argument of the logarithm (the fraction) must be equal to e raised to the power of the fraction.
Expressing the number e raised to some power, we have the equation e^(x - 1/(x² + 4)) = (x - 1)/(x² + 4). Since the base e raised to a power and the natural log are inverse operations, this equation can only be true if x - 1/(x² + 4) = 1, indicating that x = 1 + 1/(x² + 4).
To solve, we would typically proceed to clear the fraction by multiplying through by (x² + 4), and then solve the resulting quadratic equation for x.