57.  The value of log_𝑛 4⁻¹⁶ is −32. The value of 𝑛 is            .

Find the Value of n if logₙ(4⁻¹⁶) = −32

Correct Answer

✅ Correct Answer: 2

Understanding the Logarithmic Equation

The fundamental definition of logarithms states that if

log_a(b) = c

then it can be written in exponential form as

a^c = b

This relationship allows logarithmic equations to be solved by converting them into equivalent exponential equations.

Step-by-Step Solution

The given equation is:

log_n(4⁻¹⁶) = −32

Using the definition of logarithms,

n⁻³² = 4⁻¹⁶

Express 4 as a power of 2.

4 = 2²

Therefore,

4⁻¹⁶ = (2²)⁻¹⁶ = 2⁻³²

The equation now becomes:

n⁻³² = 2⁻³²

Since the exponents are equal on both sides, the bases must also be equal.

n = 2

Verification

Substitute n = 2 into the original equation.

log₂(4⁻¹⁶)

Since

4⁻¹⁶ = 2⁻³²

Therefore,

log₂(2⁻³²) = −32

The given equation is satisfied, confirming that the calculated value is correct.

Concept Behind the Question

This problem tests the basic definition of logarithms and the laws of exponents. Students should remember that every logarithmic equation can be transformed into an equivalent exponential equation. After conversion, simplifying powers often makes the solution straightforward.

Final Answer

Converting the logarithmic equation into exponential form and simplifying the powers gives:

n = 2

✅ Correct Answer: 2