91Ó°ÊÓ

To solve logarithmic equations involving radicals, we first need to convert the radical into its exponential form. This makes it easier to work with the equation. The radical symbol ( \sqrt{} ) is commonly used to represent root values.
For example, \( \sqrt[4]{8} \), the fourth root of 8, can be converted to an exponent form. It is equivalent to raising 8 to the power of \(\frac{1}{4}\). So, we have:
\[ \sqrt[4]{8} = 8^{1/4} \].
With this conversion, we simplify our initial problem significantly and prepare it for logarithmic manipulation.

Logarithmic properties are powerful tools that help simplify and solve equations involving logarithms. One such property states: \( \text{log}_a(b^c) = c \text{log}_a(b) \).
This property allows us to move the exponent in an expression to the front as a multiplier. For our problem, we have \( \log_{8}(8^{1/4}) \). Using the property, we can rewrite it as:
\[ \log_{8}(8^{1/4}) = \frac{1}{4} \text{log}_{8}(8) \].
This step simplifies the logarithmic equation considerably by reducing the complexity of the expression.

The final part involves simplifying the logarithmic expression. One basic rule of logarithms is that the log of a number to its own base is 1. In simpler terms: \( \log_{b}(b) = 1 \).
Applying this to our equation, we have \( \log_{8}(8)\) which equals 1.
Substituting this value back into our rewritten equation from the previous step, we get:
\[ \frac{1}{4} \log_{8}(8) = \frac{1}{4} \times 1 = \frac{1}{4} \].
As a result, our original logarithmic equation \( x = \log_{8} \sqrt[4]{8} \) simplifies to \( x = \frac{1}{4} \). This solution not only showcases the transformation of a radical to an exponent and the critical application of logarithmic properties but also emphasizes the importance of logarithmic simplification in problem-solving.