Math Problem Statement

Find the logarithm using the change of base formula. log Subscript 3 Baseline 19 . . . Question content area right Part 1 log Subscript 3 Baseline 19equals    enter your response here ​(Round to four decimal places as​ needed.)

Solution

To find log319\log_3 19 using the change of base formula, we convert it to a base-10 (or base-ee) logarithm. The change of base formula is:

log319=log19log3\log_3 19 = \frac{\log 19}{\log 3}

  1. Calculate log19\log 19 and log3\log 3:

    • log191.2788\log 19 \approx 1.2788
    • log30.4771\log 3 \approx 0.4771
  2. Substitute these values into the formula:

log3191.27880.47712.6799\log_3 19 \approx \frac{1.2788}{0.4771} \approx 2.6799

Thus:

log3192.6799\log_3 19 \approx 2.6799

Final Answer:

log3192.6799\log_3 19 \approx 2.6799

Would you like a more detailed breakdown or have any questions?


Related Questions:

  1. How does the change of base formula work for other logarithmic bases?
  2. What is the natural logarithm ln\ln equivalent of log319\log_3 19?
  3. How would you find log550\log_5 50 using the same method?
  4. Why is the change of base formula useful in logarithmic calculations?
  5. What is the result of log319\log_3 19 using base-ee instead of base-10?

Tip: Remember, using the change of base formula can simplify calculations when a calculator only provides log\log (base-10) or ln\ln (base-ee).

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Math Problem Analysis

Mathematical Concepts

Logarithms
Change of Base Formula

Formulas

Change of Base Formula: log_b(a) = log_c(a) / log_c(b)

Theorems

Properties of Logarithms

Suitable Grade Level

Grades 9-12