To solve the equation \( \log_{1/16} (16^x) = \frac{1}{4} \), let's proceed with the following steps:

### Step 1: Express the logarithm
We know that if \( \log_a (b) = c \), then \( b = a^c \). So, applying this property:

\[
\log_{1/16} (16^x) = \frac{1}{4} \quad \Rightarrow \quad 16^x = (1/16)^{1/4}
\]

### Step 2: Simplify the expression on the right-hand side
The term \( (1/16)^{1/4} \) can be rewritten as:

\[
(1/16)^{1/4} = \left(\frac{1}{2^4}\right)^{1/4} = \frac{1}{2} = 2^{-1}
\]

So, the equation becomes:

\[
16^x = 2^{-1}
\]

### Step 3: Express \(16\) as a power of \(2\)
Since \( 16 = 2^4 \), we can rewrite the equation as:

\[
(2^4)^x = 2^{-1}
\]

This simplifies to:

\[
2^{4x} = 2^{-1}
\]

### Step 4: Solve for \(x\)
Since the bases are the same, we can equate the exponents:

\[
4x = -1
\]

Dividing both sides by 4:

\[
x = -\frac{1}{4}
\]

### Final Answer:
\[
x = -\frac{1}{4}
\]

Would you like more details on this solution or have any questions? 

Here are some related questions:

1. How do logarithms work with different bases?
2. What are some properties of logarithms that simplify complex equations?
3. How can logarithmic identities be used in solving exponential equations?
4. What are common mistakes to avoid when working with logarithmic equations?
5. How do you solve equations involving multiple logarithmic terms?
6. What happens when the logarithm's base is a fraction, as in this problem?
7. How does changing the base of a logarithm affect the equation's solution?
8. Can you solve a logarithmic equation using a calculator for more complicated bases?

**Tip:** When solving logarithmic equations, it’s crucial to remember the property that \( \log_a(b) = c \) implies \( b = a^c \), which often helps to convert the logarithm into an exponential form for easier manipulation.

Learn how to solve the logarithmic equation log1/16(16^x) = 1/4 with a step-by-step explanation. This problem involves logarithmic identities and exponential equations. Suitable for high school students in Grades 11-12.

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