Final answer:
To find log_8(500), we use the change of base formula and a calculator to determine that log_8(500) is approximately 2.989 after rounding to three decimal places.
Step-by-step explanation:
To evaluate the logarithm log8(500), we can use the change of base formula which allows us to express this logarithm in terms of common logarithms or natural logarithms that can be calculated using a calculator:
The change of base formula is given by logb(a) = logc(a) / logc(b),
where c is any positive value, and for our purposes we can take c to be 10 to use the common logarithm.
Therefore, log8(500) = log(500) / log(8).
By using a calculator, we find that log(500) ≈ 2.6990 and log(8) ≈ 0.9031.
Thus, dividing these two values:
log8(500) ≈ 2.6990 / 0.9031
≈ 2.989 (rounded to three decimal places to match the question's requirement).