Use a change of base formula to evaluate the logarithm. Give your answer rounded to four decimal places.
[tex]log _3 \sqrt{3}[/tex]
[tex]log _3 \sqrt{3}=\square[/tex]
(Simplify your answer. Do not round until the final answer. Then round to four decimal places as needed.)
Answer (1)
Rewrite 3 as 3 2 1 .
Use the property of logarithms: lo g a a x = x .
Therefore, lo g 3 3 = lo g 3 3 2 1 = 2 1 .
Convert 2 1 to a decimal and round to four decimal places: 0.5000 .
Explanation
Understanding the problem We are asked to evaluate the logarithm lo g 3 3 and round the answer to four decimal places.
Rewriting the square root First, we can rewrite the square root of 3 as 3 raised to the power of 2 1 . So, we have 3 = 3 2 1 .
Substituting the rewritten square root Now we can rewrite the original expression as: lo g 3 3 = lo g 3 3 2 1 .
Applying the logarithm property Using the property of logarithms that lo g a a x = x , we have: lo g 3 3 2 1 = 2 1 .
Converting to decimal Now, we convert the fraction 2 1 to a decimal: 2 1 = 0.5 .
Rounding to four decimal places Finally, we round the decimal to four decimal places. Since 0.5 has only one decimal place, we can add three zeros to the end without changing its value: 0.5 = 0.5000 .
Final Answer Therefore, the final answer is 0.5000.
Examples
Logarithms are used in many real-world applications, such as measuring the intensity of earthquakes (the Richter scale), the loudness of sound (decibels), and the acidity of a solution (pH). Understanding logarithms helps us to work with quantities that vary over a very large range, making them easier to comprehend and compare. For example, the Richter scale uses logarithms to represent earthquake magnitudes, where each whole number increase represents a tenfold increase in amplitude. This allows us to express a wide range of earthquake intensities in a manageable scale.
