Final answer:
To solve the logarithmic problem log₈x=-2, the exponential relationship is applied, revealing that x equals 1/64.
Step-by-step explanation:
The logarithmic problem given by log₈x=-2 is asking us to find the value of x when the base-8 logarithm of x equals -2. To solve this, we use the definition of logarithms: If log₈x = y, then 8^y = x. Applying this definition, we have 8^-2 = x.
The exponential and logarithmic functions are inverse to each other. This means that taking the natural logarithm of an exponent will cancel out, leaving the exponent itself (as in ln(e^x) = x), and raising e to the power of a natural logarithm will also cancel out, leaving the argument of the logarithm (as in e^(ln x) = x). This trick is helpful for understanding the relationship between logarithms and exponents.
Now, knowing that 8^-2 = x, we can simplify this to x = 1/8^2, which equals 1/64. Therefore, the value of x is 1/64.