Final answer:
The expression equivalent to log₃₂₇ is c) ln27/ln3. It is derived from the change of base formula and the property that the logarithm of a number raised to an exponent is the product of the exponent and the logarithm of the number. Therefore, the correct answer is option c) ln27ln3
Step-by-step explanation:
Understanding Logarithms
When considering which expressions are equivalent to log₃₂₇, it is essential to know certain properties of logarithms. Specifically, one property states that the logarithm of a number raised to an exponent is the product of the exponent and the logarithm of the number. Therefore, we can rewrite log₃₂₇ as 3 × log₃(3), because 27 is 3 to the power of 3.
Let's evaluate the given options:
- a) log₃log₂₇ : This expression does not apply, as it suggests taking the logarithm of another logarithm rather than addressing the exponent.
- b) log₂₇log₃ : This expression is also incorrect because it implies multiplication between two logarithms with different bases, which is not equivalent to the given expression.
- c) ln27/ln3 : Considering the change of base formula, this is correct. Since the natural logarithm can be used with any base, the expression simplifies to log₃₂₇.
- d) log₂₇³ : This is incorrect as it suggests the logarithm of 27 to the power of 3, which is not the same as the given expression.
Therefore, the only equivalent expression to log₃₂₇ is c) ln27/ln3.