Solve 16$^x$ = 0.25

Analysis

The equation given is an exponential equation: $16^x = 0.25$. To solve for $x$, we need to express both sides of the equation using the same base.

Concept: Laws of Indices

1. Expressing decimals as fractions: $0.25$ is the same as $\frac{25}{100}$, which simplifies to $\frac{1}{4}$.
2. Negative Indices: Recall that $\frac{1}{a^n} = a^{-n}$.
3. Common Base: Both $16$ and $4$ are powers of $2$. Specifically, $16 = 2^4$ and $4 = 2^2$.

Step-by-Step Solution

Step 1: Rewrite the equation with fractions

Step 2: Express both sides using base 2 Since $16 = 2^4$:

Step 3: Apply the laws of indices On the left side, use the power of a power rule $(a^m)^n = a^{m \times n}$:

On the right side, use the negative index rule $\frac{1}{a^n} = a^{-n}$:

Step 4: Equate the exponents Since the bases are now the same ($2$), we can equate the exponents:

Step 5: Solve for $x$ Divide both sides by 4:

Conclusion

Comparing our result with the given options, the correct value is $-\frac{1}{2}$.

Exam Tip

Whenever you see a decimal like $0.25$, $0.5$, or $0.125$ in an exponential equation, immediately convert it to a fraction ($\frac{1}{4}$, $\frac{1}{2}$, or $\frac{1}{8}$). It makes it much easier to spot the common base!