Solve the Product: log₄6 × log₆9 × log₉4 Logarithm Chain

$\log_46\times\log_69\times\log_94=$

To solve this problem, we need to recognize that the expression $\log_46 \times \log_69 \times \log_94$ fits the identity of logarithms: $\log_a b \times \log_b c \times \log_c a = 1$.

Let us examine the expression:

• The first term is $\log_46$, where $a = 4$ and $b = 6$.
• The second term is $\log_69$, where $b = 6$ and $c = 9$.
• The third term is $\log_94$, where $c = 9$ and $a = 4$.

Notice how $\log_46$, $\log_69$, and $\log_94$ correspond respectively to $\log_a b$, $\log_b c$, and $\log_c a$. Thus, the entire expression matches the multiplication identity of logarithms: $\log_a b \times \log_b c \times \log_c a = 1$.

Therefore, the value of the expression $\log_46 \times \log_69 \times \log_94$ is $1$.