How can you simplify the expression 7 log3 k + 6 log3 m + 9 log3 n?

To simplify the expression 7 log₃ k + 6 log₃ m + 9 log₃ n, we can make use of the properties of logarithms. The key properties we will use are:

• Product Rule: log_b (xy) = log_b x + log_b y
• Power Rule: log_b (xⁿ) = n log_b x

First, we can apply the Power Rule to each term in the expression:

• 7 log₃ k can be rewritten as log₃ (k⁷)
• 6 log₃ m can be rewritten as log₃ (m⁶)
• 9 log₃ n can be rewritten as log₃ (n⁹)

Now, substituting these back into our expression gives us:

log₃ (k⁷) + log₃ (m⁶) + log₃ (n⁹)

By applying the Product Rule for logarithms, we can combine these terms into a single logarithm:

log₃ (k⁷ m⁶ n⁹)

Thus, the simplified form of the original expression 7 log₃ k + 6 log₃ m + 9 log₃ n is:

log₃ (k⁷ m⁶ n⁹)

This expression can be useful in many contexts, particularly in fields such as mathematics and computer science, where logarithmic calculations are frequently employed.