Final answer:
The value of log8(x) in the given equation is 1, after simplifying the given logarithmic expressions and solving for x using properties of logarithms and exponential functions.
Step-by-step explanation:
To solve the equation log8(x²) + log64(x) = 8, we need to simplify and solve for x. Since both logarithms have different bases, we will convert them to a common base. The base of the second logarithm, 64, is 8 squared (8² = 64). Knowing this, we apply the change of base formula which is logb(x) = logk(x) / logk(b), where k is a new base.
So, log64(x) can be written as log8(x) / log8(64), which simplifies to log8(x) / 2. When we substitute this back into the original equation, we get log8(x²) + 0.5 * log8(x) = 8.
Using the properties of logarithms, we combine the terms: log8(x³) = 8. Now we'll use the definition of logarithm, which tells us that if logb(x) = y, then b&supy; = x. Thus, we have 8³ = x³, and taking the cube root of both sides gives us x = 8.
Finally, we can evaluate log8(x), which is log8(8) and equals 1 since any log base b of b equals 1.