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To solve the logarithmic expression using the change-of-base formula, let's follow these steps:

### Expression:
\[
\log_{\sqrt{19}}{3}
\]

### Step 1: Change-of-base formula
The change-of-base formula for logarithms is:
\[
\log_b{a} = \frac{\log_c{a}}{\log_c{b}}
\]
Where \( b \) is the base and \( a \) is the argument. Here \( b = \sqrt{19} \) and \( a = 3 \).

Using this, we get:
\[
\log_{\sqrt{19}}{3} = \frac{\log{3}}{\log{\sqrt{19}}}
\]

### Step 2: Simplify the base
Since \( \log{\sqrt{19}} = \log{19^{1/2}} = \frac{1}{2} \log{19} \), we can rewrite the expression as:
\[
\log_{\sqrt{19}}{3} = \frac{\log{3}}{\frac{1}{2} \log{19}} = \frac{2 \log{3}}{\log{19}}
\]

### Step 3: Calculate the values using a calculator
We can now use approximate values for the logarithms (base 10):

- \( \log{3} \approx 0.4771 \)
- \( \log{19} \approx 1.2788 \)

Substitute these values:
\[
\log_{\sqrt{19}}{3} = \frac{2 \times 0.4771}{1.2788} = \frac{0.9542}{1.2788} \approx 0.7462
\]

### Final Answer:
\[
\log_{\sqrt{19}}{3} \approx 0.746
\]

Do you have any questions, or would you like further details on the process?

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Here are 5 related questions for practice:
1. How would you solve \( \log_{\sqrt{25}}{5} \)?
2. Can you simplify \( \log_{25}{5} \) without the change-of-base formula?
3. What is the value of \( \log_3{81} \)?
4. How do you apply the change-of-base formula to logarithms with a base \( e \)?
5. How does the logarithmic property \( \log_b{(x^n)} = n \log_b{x} \) apply in this problem?

**Tip:** When solving logarithmic problems, remember that simplifying the base using exponent rules often makes the calculation much easier.

Use the change-of-base formula and a calculator to evaluate each logarithm. Round your answer to three decimal places: log_√19 (3)

Learn how to use the change-of-base formula to evaluate the logarithm log_√19 (3). This problem involves key concepts like the change-of-base formula and exponent properties. Suitable for students in Grades 9-12.

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