Question

Which expression is equivalent to ( log_4 8 )?

a) ( (log_4 8)³ )
b) ( log_4 3 )
c) ( 8 log_4 8 )
d) ( 3 log_4 8 )

Answer 1

The expression equivalent to ( log_4 8 ) is ( 3 log_4 8 ) because 8 is 2 cubed, and 2 is the base of 4 squared, which means log_4 8 simplifies to 3 times log_4 2, and log_4 2 is 1. Therefore, the correct answer is option d).

Step-by-step explanation:

The student has asked which expression is equivalent to ( log_4 8 ). We can use the property of logarithms which states that the logarithm of a number raised to an exponent is the product of the exponent and the logarithm of the number. We can easily evaluate ( log_4 8 ) because 8 is 2 cubed (23), and 2 is the square of 4, the base of our logarithm (4 = 22). Therefore:

log_4 8 = log_4 (23)

Using property of logarithms = 3 × log_4 2

Since 4 is 2 squared, log_4 2 = 1

Thus, 3 × 1 = 3

The equivalent expression is therefore ( 3 log_4 8 ), which is option d). It is important to recognize that the base of the logarithm and the number being logged have a power relationship that simplifies the expression.