What is the value of $$ \log _{7} \sqrt[4]{7} $$ ?

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UpStudy AI · Accepted Answer

To find the value of $$ \log_{7} \sqrt[4]{7} $$, let's break it down step by step.

### Step 1: Express the Fourth Root as an Exponent
The fourth root of 7 can be written as an exponent:
$$
\sqrt[4]{7} = 7^{1/4}
$$

### Step 2: Apply the Logarithm Power Rule
The logarithm power rule states that:
$$
\log_b (a^c) = c \cdot \log_b a
$$
Applying this rule to $$ \log_{7} 7^{1/4} $$:
$$
\log_{7} 7^{1/4} = \frac{1}{4} \cdot \log_{7} 7
$$

### Step 3: Evaluate $$ \log_{7} 7 $$
Since the logarithm of a number to its own base is 1:
$$
\log_{7} 7 = 1
$$

### Step 4: Combine the Results
Substituting back:
$$
\frac{1}{4} \cdot 1 = \frac{1}{4}
$$

### Final Answer
$$
\log_{7} \sqrt[4]{7} = \frac{1}{4}
$$
The value of $$ \log_{7} \sqrt[4]{7} $$ is $$ \frac{1}{4} $$.

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