Given [tex]log _4 3 \approx 0.792[/tex] and [tex]log _4 21 \approx 2.196[/tex], what is [tex]log _4 7[/tex]?
Answer (1)
Use the logarithm property lo g a ( b c ) = lo g a b + lo g a c to express lo g 4 21 as lo g 4 3 + lo g 4 7 .
Rearrange the equation to isolate lo g 4 7 : lo g 4 7 = lo g 4 21 − lo g 4 3 .
Substitute the given values: lo g 4 7 = 2.196 − 0.792 .
Calculate the result: lo g 4 7 = 1.404 .
Explanation
Problem Analysis We are given that lo g 4 3 ≈ 0.792 and lo g 4 21 ≈ 2.196 . We want to find the value of lo g 4 7 .
Using Logarithm Properties We know that 21 = 3 × 7 . Using the logarithm property lo g a ( x y ) = lo g a x + lo g a y , we can write lo g 4 21 = lo g 4 ( 3 × 7 ) = lo g 4 3 + lo g 4 7 .
Isolating the Desired Term Now, we can solve for lo g 4 7 by rearranging the equation: lo g 4 7 = lo g 4 21 − lo g 4 3 .
Substituting Values Substitute the given values: lo g 4 7 = 2.196 − 0.792 .
Calculating the Result Calculating the result, we get lo g 4 7 = 1.404 .
Final Answer Therefore, lo g 4 7 ≈ 1.404 .
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 formula is M = lo g 10 ( A ) , where M is the magnitude and A is the amplitude of the seismic waves. Logarithms also appear in calculating the pH of a solution in chemistry, measuring sound intensity in decibels, and in various financial calculations involving exponential growth or decay. Understanding logarithms helps us quantify and analyze phenomena that vary over a wide range of values.
