Final answer:
The correct logarithmic form of the equation x⁴ = 2 is 4ln(x) = ln(2), obtained by taking the natural log of both sides and using the property that ln(xⁿ) = f * ln(x).
"the correct option is approximately option A"
Step-by-step explanation:
To rewrite the equation x⁴ = 2 in logarithmic form using the natural logarithm, we apply the property that the logarithm of a number raised to an exponent is the product of the exponent and the logarithm of the base number. The natural logarithm, often denoted as ln, is the amount of time needed to reach a certain level of continuous growth and is the inverse function of the exponential function with base e (Euler's number, approximately 2.7182818). Therefore, applying the logarithmic property - if bⁿ = x, then f = ln(x)/ln(b) - we convert the exponential equation into its logarithmic counterpart.
Starting with the equation x⁴ = 2, we take the natural logarithm of both sides:
ln(x⁴) = ln(2)
Using the property that ln(xⁿ) = f * ln(x), where f is the exponent, we can rewrite the equation as:
4 * ln(x) = ln(2)
Therefore, the correct logarithmic form of the given equation is 4ln(x) = ln(2), not 4ln(x) = 2. This is because the equation originally states that x raised to the power of four equals two, and thus we take the natural log of two, not state that it equals two directly.