Final answer:
To evaluate log₈ 729, determine the power to which 8 must be raised to obtain 729 using linear interpolation. The value of log₈ 729 is approximately 3.815. To evaluate log₀ 7776 - log₆ 36, rewrite the logarithms in a common base (such as base 10). The value of log₀ 7776 is undefined, while log₆ 36 is approximately 2. The result is -∞.
Step-by-step explanation:
To evaluate log₈ 729, we need to determine the power to which 8 must be raised in order to obtain 729. In this case, 8³ = 512 and 8⁴ = 4096. Since 729 is between these two values, we can conclude that log₈ 729 lies between 3 and 4.
Using linear interpolation, we can find the exact value. The difference between 729 and 512 is 217, and the difference between 4096 and 512 is 3584. The difference between 3 and 4 is 1. By setting up a proportion, we find that (1/217) = (1/3584)(x-3), where x represents the value of log₈ 729. Solving for x gives us x ≈ 3.815.
Therefore, log₈ 729 ≈ 3.815.
To evaluate log₀ 7776 - log₆ 36, we first need to express both logarithms in a common base. Since logarithms with a base of 0 are undefined, we'll use a different base to rewrite the logarithm. Let's use base 10. Converting log₀ 7776 to base 10 gives us: log₀ 7776 = log₁₀ 7776 / log₁₀ 0 = log₁₀ 7776 / -∞ = -∞.
Next, we convert log₆ 36 to base 10: log₆ 36 = log₁₀ 36 / log₁₀ 6. Using a calculator, we find that log₁₀ 36 ≈ 1.556 and log₁₀ 6 ≈ 0.778. Therefore, log₆ 36 ≈ 1.556 / 0.778 ≈ 2.
Finally, we can substitute the values into the equation: log₋ 7776 - log₆ 36 = -∞ - 2 = -∞.