4.4 (11) laws of logarithms

Question

Use the Laws of Logarithms to combine the expression.

(1/2)log₂(6) − 2 log₂(7)

Answer 1

There are three main properties of logarithms to use in questions like this:

Product property: log_b(MN) = log_bM + log_bN

Quotient property: log_b(M / N) = log_bM - log_bN

Power property: log_b(M^N) = N•log_bM

NOTE: these properties can be used in either direction.

For your specific example, you will need to use the Quotient and Power properties only (there is no addition of terms anywhere), applying the properties from right to left. Following order of operations, you would use Power property first to get:

log₂6^1/2 - log₂7²

Quotient property next:

log₂(6^1/2 / 7²)

which could also be written as log₂(√6 / 49)

Answer 2

(1/2)log₂(6) − 2 log₂(7)

The first step is to take the number in front and move it to the exponent

log₂(6)^1/2 − log₂(7)²

Next, subtraction correlates to division. The next step would be

log₂(6^1/2 /7²)

Simplifying what can be simplified:

log₂(6^1/2 /49)

Answer 3

There a probably several ways to answer this question:

one way is log₂(6^1/2/7²).