Given [tex]log _3 2 \approx 0.631[/tex] and [tex]log _3 7 \approx 1.771[/tex], what is [tex]log _3 14[/tex]?
Answer (2)
Rewrite 14 as a product of 2 and 7: 14 = 2 × 7 .
Apply the logarithm product rule: lo g 3 14 = lo g 3 2 + lo g 3 7 .
Substitute the given approximations: lo g 3 14 ≈ 0.631 + 1.771 .
Calculate the sum: lo g 3 14 ≈ 2.402 .
Explanation
Understanding the Problem We are given that lo g 3 2 ≈ 0.631 and lo g 3 7 ≈ 1.771 . We want to find lo g 3 14 .
Rewriting the Expression We can rewrite 14 as 2 ⋅ 7 . Therefore, lo g 3 14 = lo g 3 ( 2 ⋅ 7 ) .
Applying Logarithm Properties Using the property of logarithms that lo g b ( mn ) = lo g b m + lo g b n , we have lo g 3 ( 2 ⋅ 7 ) = lo g 3 2 + lo g 3 7 .
Substituting Values Substituting the given values, we get lo g 3 14 ≈ 0.631 + 1.771 .
Calculating the Sum Calculating the sum, we find that lo g 3 14 ≈ 2.402 .
Final Answer Therefore, lo g 3 14 ≈ 2.402 .
Examples
Logarithms are incredibly useful in many real-world scenarios. For example, they are used to measure the magnitude of earthquakes on the Richter scale. The Richter scale is logarithmic, meaning that each whole number increase on the scale represents a tenfold increase in the amplitude of the earthquake waves. Similarly, logarithms are used in chemistry to measure pH levels, in acoustics to measure sound intensity (decibels), and in finance to calculate compound interest. Understanding logarithms helps us to quantify and compare phenomena that vary over a wide range of values.
To find lo g 3 14 , we rewrite 14 as 2 × 7 and apply the logarithm product rule. By substituting the given values, we calculate lo g 3 14 ≈ 2.402 .
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