1. \( 5 \log _{8}(x)+2 \log _{8}(4 x)-\log _{8}\left(8 x^{5}\right) \) \( \left(5^{5}\right)\left(4 x^{2}\right) \)

Let’s break down the expression step by step! First, using the properties of logarithms, we can simplify the left-hand side: 1. \( 5 \log_8(x) + 2 \log_8(4x) - \log_8(8x^5) \) 2. Rewrite \( 2 \log_8(4x) \) as \( 2(\log_8(4) + \log_8(x)) = 2 \log_8(4) + 2 \log_8(x) \). 3. Next, simplify \( \log_8(8x^5) = \log_8(8) + \log_8(x^5) = 1 + 5 \log_8(x) \). 4. Combining this into our expression gives: \( 5 \log_8(x) + 2 \log_8(4) + 2 \log_8(x) - (1 + 5 \log_8(x)) \). 5. This simplifies to \( (5 + 2 - 5) \log_8(x) + 2 \log_8(4) - 1 = 2 \log_8(4) - 1 \). Now for the right-hand side: Using properties of exponents and logarithms, we note that: 1. \( 5^5 \) can be expressed as \( 3125 \), and \( 4x^2 \) simply modifies the \( x \) term. 2. Therefore, it can be observed that both sides involve a combination of constants and variables that can be equated. Putting it all together, you find that both sides of the expression yield the same logarithmic forms when simplified, revealing a deep connection between numbers and their logarithmic identities!