To evaluate the given expression $\log_{10} \sqrt[4]{10}$, follow these steps:

Step 1: Simplify the argument of the logarithm

The fourth root of 10 can be rewritten as: $\sqrt[4]{10} = 10^{1/4}$

Step 2: Use the logarithmic property

The logarithmic property states: $\log_b(a^c) = c \cdot \log_b(a)$ Here, $a = 10$, $b = 10$, and $c = \frac{1}{4}$. Substituting into the property gives: $\log_{10}(10^{1/4}) = \frac{1}{4} \cdot \log_{10}(10)$

Step 3: Evaluate $\log_{10}(10)$

By definition of logarithms: $\log_{10}(10) = 1$

Step 4: Simplify the result

Substitute $\log_{10}(10) = 1$ into the expression: $\frac{1}{4} \cdot 1 = \frac{1}{4}$

Final Answer:

$\log_{10} \sqrt[4]{10} = \frac{1}{4}$

Let me know if you'd like further clarification or additional examples! Here are some related questions to consider:

1. What are the general properties of logarithms and how can they simplify calculations?
2. How does changing the base of the logarithm affect the result?
3. What is the relationship between roots and fractional exponents?
4. Can you extend this result to $\log_b \sqrt[4]{b^n}$?
5. How do logarithms apply in real-world situations such as pH or decibels?

Tip: Always remember that roots can be expressed as fractional exponents, which makes applying logarithmic rules much easier!