If [tex]log_{10} 8 = 0.9031[/tex], find the value of [tex]4 \times log_{10} 16[/tex].
Answer (2)
Express 16 and 8 as powers of 2.
Use the logarithm property lo g b a c = c lo g b a to rewrite the expression.
Calculate lo g 10 2 using the given information.
Substitute the value to find 4 lo g 10 16 = 16 × 3 0.9031 ≈ 4.8165 .
The final answer is 4.8165 .
Explanation
Problem Setup We are given that lo g 10 8 = 0.9031 and we want to find the value of 4 lo g 10 16 .
Expressing as Powers of 2 First, we can express 8 and 16 as powers of 2. We have 8 = 2 3 and 16 = 2 4 .
Using Logarithm Properties Using the logarithm property lo g b a c = c lo g b a , we can rewrite the given information as lo g 10 8 = lo g 10 2 3 = 3 lo g 10 2 = 0.9031 .
Finding log base 10 of 2 From this, we can find lo g 10 2 by dividing both sides by 3: lo g 10 2 = 3 0.9031 .
Rewriting log base 10 of 16 Now, we can rewrite lo g 10 16 as lo g 10 2 4 = 4 lo g 10 2 .
Substituting and Simplifying We want to find 4 lo g 10 16 , which can be written as 4 ( 4 lo g 10 2 ) = 16 lo g 10 2 .
Final Calculation Substituting the value of lo g 10 2 we found earlier, we get 16 lo g 10 2 = 16 × 3 0.9031 .
Result Calculating this value, we get 16 × 3 0.9031 = 3 14.4496 = 4.81653333... ≈ 4.8165 .
Final Answer Therefore, the value of 4 lo g 10 16 is approximately 4.8165 .
Examples
Logarithms are incredibly useful in many real-world scenarios, especially when dealing with exponential growth or decay. For instance, calculating the magnitude of earthquakes using the Richter scale involves logarithms. If an earthquake is 100 times more intense than another, the Richter scale value increases by 2 (since lo g 10 100 = 2 ). Similarly, in finance, understanding compound interest involves logarithmic calculations to determine how investments grow over time. The concept of decibels, used to measure sound intensity, also relies on logarithms, allowing us to represent a wide range of sound levels on a manageable scale.
To find 4 × lo g 10 16 given lo g 10 8 = 0.9031 , we utilize properties of logarithms and express the values in terms of powers of 2. After calculating lo g 10 2 , we find that 4 × lo g 10 16 ≈ 4.8165 .
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