Log Base 4 of 64

What is log base 4 of 64?
log₄(64) = 3
3 is the log base 4 of 64.

Use in Mathematics: Log Base 4 of 64
The below are some of the mathematical applications where log base 4 of 64 can be used:

1. to solve exponential equations where the unknown exponent y satisfies 4^y = 64.
2. to simplify complex multiplication or division operations into addition or subtraction of logarithms.
3. to measure logarithmic scale changes, time complexity of algorithms, and calculus functions.

1. By logarithmic definition, log_b(x) = y means b^y = x. Here, 4³ = 64.
2. Change of base rule allows converting to natural log (ln) or common log (log₁₀): log₄(64) = ln(64) / ln(4) = log₁₀(64) / log₁₀(4) = 3.
3. Logarithms are defined only for a positive base (b > 0, b ≠ 1) and a positive argument (x > 0).

This step-by-step solution is one of the simple methods to easily understand how to find what is the Log Base 4 of 64, provided along with the use cases of mathematics and real-world problems.

Formula: log₄ 64
log_b(x) = y, if b^y = x

Input:
x = 64
b = 4

step 1 Find the prime factors for 64:
Express 64 in terms of base 4: 64 = 4³
log₄(64) = log₄(4³)

step 2 Apply logarithm property of multiplication/power:
log₄(64) = log₄(4³) = 3 log₄(4)
= 3 log₄(4)

step 3 Use the log table to find and apply the log values:
= 3 x 1

step 4 Find the sum:
= 3
log₄(64) = 3