Michael was asked to properly evaluate this logarithm. Which answer should he come to?

-9

Ask by Page Mills. in the United States

Jan 25,2025

Question

Michael was asked to properly evaluate this logarithm. Which answer should he come to? $$ \log _{9} 1 / 2 $$ - $$ -1 / 2 $$ -9

UpStudy AI · Accepted Answer

To evaluate the logarithm $$ \log_{9} \left( \frac{1}{2} \right) $$, we need to determine the exponent $$ x $$ such that:

$$
9^x = \frac{1}{2}
$$

Let's analyze the given options:

1. **Option 1: $$ x = -\frac{1}{2} $$**

Calculate $$ 9^{-1/2} $$:

$$
   9^{-1/2} = \frac{1}{\sqrt{9}} = \frac{1}{3}
   $$

This does not equal $$ \frac{1}{2} $$.

2. **Option 2: $$ x = -9 $$**

Calculate $$ 9^{-9} $$:

$$
   9^{-9} = \frac{1}{9^9}
   $$

This is an extremely small number, far less than $$ \frac{1}{2} $$.

Neither of the provided options exactly equals $$ \log_{9} \left( \frac{1}{2} \right) $$. However, if we were to approximate or consider the closest option, $$ -\frac{1}{2} $$ is closer in magnitude to the actual value compared to $$ -9 $$.

**Actual Calculation:**

For a more precise value, we can use the change of base formula:

$$
\log_{9} \left( \frac{1}{2} \right) = \frac{\log \left( \frac{1}{2} \right)}{\log 9} = \frac{-\log 2}{2\log 3} \approx -0.315
$$

This value is closer to $$ -\frac{1}{2} $$ than to $$ -9 $$.

**Conclusion:**

Given the options, **$$ -\frac{1}{2} $$** is the more appropriate choice.

**Answer:**  
Michael should choose the answer $$ -\frac{1}{2} $$.
Michael should choose $$ -\frac{1}{2} $$ as the correct answer for $$ \log_{9} \left( \frac{1}{2} \right) $$.

Michael was asked to properly evaluate this logarithm. Which answer should he come to?

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