$$ \left. \begin{array} { l l } { 16 = 4 ^ { 2 } } \ { 8 ^ { - 2 } = \frac { 1 } { 64 } } & { 0 = \log _ { 6 } 1 } \ { \log _ { x } x ^ { 4 } = 4 } \ { m ^ { n } = P } & { \log _ { 6 } ( \frac { 1 } { 36 } ) = - 2 } \ { 3 ^ { 0 } = 1 } & { 2 = \log _ { \frac { 1 } { 4 } } x } \ { \frac { 1 } { 5 imes - 3 } = 0,001 } & { } \end{array} ight. $$

Question

$$ \left. \begin{array} { l l } { 16 = 4 ^ { 2 } } \ { 8 ^ { - 2 } = \frac { 1 } { 64 } } & { 0 = \log _ { 6 } 1 } \ { \log _ { x } x ^ { 4 } = 4 } \ { m ^ { n } = P } & { \log _ { 6 } ( \frac { 1 } { 36 } ) = - 2 } \ { 3 ^ { 0 } = 1 } & { 2 = \log _ { \frac { 1 } { 4 } } x } \ { \frac { 1 } { 5 	imes - 3 } = 0,001 } & { } \end{array} ight. $$

UpStudy AI · Accepted Answer

Let's analyze the given equations step by step.

1. **Equation 1:** $$ 16 = 4^2 $$
   - This is a true statement since $$ 4^2 = 16 $$.

2. **Equation 2:** $$ 8^{-2} = \frac{1}{64} $$
   - We can rewrite $$ 8^{-2} $$ as $$ \frac{1}{8^2} $$.
   - Since $$ 8^2 = 64 $$, we have $$ 8^{-2} = \frac{1}{64} $$, which is also true.

3. **Equation 3:** $$ 0 = \log_6 1 $$
   - The logarithm of 1 in any base is 0, so this statement is true.

4. **Equation 4:** $$ \log_x x^4 = 4 $$
   - Using the property of logarithms, we can rewrite this as $$ 4 \cdot \log_x x = 4 $$.
   - Since $$ \log_x x = 1 $$, we have $$ 4 \cdot 1 = 4 $$, which is true.

5. **Equation 5:** $$ m^n = P $$
   - This is a general exponential equation and does not provide specific values.

6. **Equation 6:** $$ \log_6 \left( \frac{1}{36} ight) = -2 $$
   - We can rewrite $$ \frac{1}{36} $$ as $$ 6^{-2} $$.
   - Therefore, $$ \log_6 (6^{-2}) = -2 $$, which is true.

7. **Equation 7:** $$ 3^0 = 1 $$
   - This is a true statement since any non-zero number raised to the power of 0 is 1.

8. **Equation 8:** $$ 2 = \log_{\frac{1}{4}} x $$
   - We can rewrite this as $$ x = \left( \frac{1}{4} ight)^2 $$.
   - Since $$ \left( \frac{1}{4} ight)^2 = \frac{1}{16} $$, we have $$ x = \frac{1}{16} $$.

9. **Equation 9:** $$ \frac{1}{5 	imes -3} = 0.001 $$
   - Calculating the left side: $$ 5 	imes -3 = -15 $$, so $$ \frac{1}{-15} $$ is approximately -0.0667, which is not equal to 0.001. This statement is false.

In summary, the true statements are:
- $$ 16 = 4^2 $$
- $$ 8^{-2} = \frac{1}{64} $$
- $$ 0 = \log_6 1 $$
- $$ \log_x x^4 = 4 $$
- $$ \log_6 \left( \frac{1}{36} ight) = -2 $$
- $$ 3^0 = 1 $$
- $$ 2 = \log_{\frac{1}{4}} x $$ (where $$ x = \frac{1}{16} $$)

The false statement is:
- $$ \frac{1}{5 	imes -3} = 0.001 $$
The true statements are: - $$ 16 = 4^2 $$ - $$ 8^{-2} = \frac{1}{64} $$ - $$ 0 = \log_6 1 $$ - $$ \log_x x^4 = 4 $$ - $$ \log_6 \left( \frac{1}{36} ight) = -2 $$ - $$ 3^0 = 1 $$ - $$ 2 = \log_{\frac{1}{4}} x $$ (where $$ x = \frac{1}{16} $$) The false statement is: - $$ \frac{1}{5 	imes -3} = 0.001 $$

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