If [tex]$\log _{10} 8=0.9031$[/tex], find the value of [tex]$4 \log _{10} 16$[/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.
Find lo g 10 2 using the given information.
Calculate 4 lo g 10 16 = 16 lo g 10 2 . The final answer is 4.8165 .
Explanation
Understanding the Problem 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 16 and 8 as powers of 2: 16 = 2 4 and 8 = 2 3 .
Applying Logarithm Properties Using the logarithm property lo g b a c = c lo g b a , we can rewrite lo g 10 16 as 4 lo g 10 2 and lo g 10 8 as 3 lo g 10 2 .
Solving for log base 10 of 2 We are given that lo g 10 8 = 0.9031 , so 3 lo g 10 2 = 0.9031 . We can solve for lo g 10 2 by dividing both sides by 3: lo g 10 2 = 3 0.9031 .
Substituting and Simplifying Now we want to find 4 lo g 10 16 . Since 16 = 2 4 , we have 4 lo g 10 16 = 4 ( 4 lo g 10 2 ) = 16 lo g 10 2 .
Calculating the Final Value Substituting the value of lo g 10 2 we found earlier, we get: 16 lo g 10 2 = 16 × 3 0.9031 = 3 16 × 0.9031 = 3 14.4496 = 4.8165333...
Final Answer Therefore, the value of 4 lo g 10 16 is approximately 4.8165 .
Examples
Logarithms are incredibly useful in various fields, such as calculating the magnitude of earthquakes on the Richter scale or determining the acidity (pH) of a solution. In finance, logarithms help in calculating investment growth rates and analyzing compound interest. For instance, if you know the initial investment, the final amount, and the time period, you can use logarithms to find the interest rate needed to achieve that growth. This makes logarithms a powerful tool for understanding exponential relationships in real-world scenarios.
To find 4 lo g 10 16 , we express 16 as 2 4 and use logarithm properties. Given that lo g 10 8 = 0.9031 , we find lo g 10 2 and subsequently calculate 4 lo g 10 16 , resulting in approximately 4.8165 .
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