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?

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.