Use common logarithms or natural logarithms and a calculator to evaluate the expression.

log Subscript 19 Baseline 12log1912

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Part 1

Evaluate the expression.

log Subscript 19 Baseline 12log1912almost equals≈enter your response here

​(Type an integer or a decimal. Do not round until the final answer. Then round to four decimal places as​ needed.)

To evaluate the expression \( \log_{19}(12) \), you can use the change of base formula. The change of base formula states:

\[
\log_{b}(a) = \frac{\log_c(a)}{\log_c(b)}
\]

where \( c \) can be any positive number (usually 10 for common logarithms or \( e \) for natural logarithms).

Let's use common logarithms (base 10):

\[
\log_{19}(12) = \frac{\log_{10}(12)}{\log_{10}(19)}
\]

Now, let's compute this using a calculator.

1. Compute \( \log_{10}(12) \).
2. Compute \( \log_{10}(19) \).
3. Divide the result from step 1 by the result from step 2.

I'll do this calculation for you.The value of \( \log_{19}(12) \) is approximately \( 0.8439 \) when rounded to four decimal places.

Would you like more details on this process or have any questions?

Here are some related questions:

1. How do you use the natural logarithm to evaluate \( \log_{19}(12) \)?
2. What is the change of base formula and when is it used?
3. How do logarithms relate to exponents?
4. What is the difference between common logarithms and natural logarithms?
5. Can you evaluate \( \log_{19}(12) \) using natural logarithms?
6. How do you solve logarithmic equations with different bases?
7. What are the properties of logarithms?
8. How do you convert logarithms with different bases?

**Tip:** Always check if your calculator is set to the correct mode (degrees/radians) when working with trigonometric functions involving logarithms.

Learn how to evaluate \( \log_{19}(12) \) using common logarithms and the change of base formula. Detailed step-by-step solution provided. Suitable for students in Grades 11-12.

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