Given $\log _3 2 \approx 0.631$ and $\log _3 7 \approx 1.771$, what is $\log _3 14$?
Answer (1)
Rewrite 14 as a product of 2 and 7: lo g 3 14 = lo g 3 ( 2 × 7 ) .
Apply the logarithm property: lo g 3 ( 2 × 7 ) = lo g 3 2 + lo g 3 7 .
Substitute the given approximations: lo g 3 2 + lo g 3 7 ≈ 0.631 + 1.771 .
Calculate the sum: 0.631 + 1.771 = 2.402 , so 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 the value of lo g 3 14 .
Rewriting the Logarithm We can rewrite 14 as the product of 2 and 7, so we have lo g 3 14 = lo g 3 ( 2 × 7 ) .
Applying the Logarithm Property Using the logarithm property lo g b ( mn ) = lo g b m + lo g b n , we can rewrite the expression as lo g 3 ( 2 × 7 ) = lo g 3 2 + lo g 3 7 .
Substituting the Approximations Now, we substitute the given approximations: lo g 3 2 + lo g 3 7 ≈ 0.631 + 1.771 .
Calculating the Sum Adding the two values, we get 0.631 + 1.771 = 2.402 . Therefore, lo g 3 14 ≈ 2.402 .
Examples
Logarithms are used in many real-world applications, such as calculating the magnitude of earthquakes on the Richter scale, measuring sound intensity in decibels, and determining the pH of a solution in chemistry. In finance, logarithms are used to calculate the time it takes for an investment to double at a certain interest rate. Understanding logarithmic properties helps in simplifying complex calculations in these fields.
