If [tex]log_{10} 8 = 0.9031[/tex], find the value of [tex]log_{10} 16[/tex].
Answer (2)
Express 16 as a power of 8: 16 = 8 3 4 .
Use the logarithm property: lo g 10 16 = 3 4 lo g 10 8 .
Substitute the given value: lo g 10 16 = 3 4 × 0.9031 .
Calculate the final value: lo g 10 16 = 1.2041 .
Explanation
Understanding the Problem We are given that lo g 10 8 = 0.9031 and we want to find the value of lo g 10 16 .
Expressing 16 and 8 as powers of 2 We can express 16 as a power of 2, i.e., 16 = 2 4 . Similarly, we can express 8 as a power of 2, i.e., 8 = 2 3 .
Expressing 16 as a power of 8 From 8 = 2 3 , we can write 2 = 8 3 1 . Therefore, 16 = 2 4 = ( 8 3 1 ) 4 = 8 3 4 .
Using Logarithm Properties Now, we can find lo g 10 16 using the properties of logarithms: lo g 10 16 = lo g 10 ( 8 3 4 ) = 3 4 lo g 10 8
Calculating the Value We are given that lo g 10 8 = 0.9031 . Substituting this value into the equation above, we get: lo g 10 16 = 3 4 × 0.9031 = 1.2041333333333333 Rounding to four decimal places, we have lo g 10 16 ≈ 1.2041 .
Examples
Logarithms are incredibly useful in many real-world applications, especially when dealing with exponential growth or decay. For instance, in finance, logarithms can help calculate the time it takes for an investment to double at a certain interest rate. In chemistry, they're used to measure pH levels, which indicate the acidity or alkalinity of a substance. In seismology, the Richter scale, which measures the magnitude of earthquakes, is logarithmic, allowing us to compare the energy released by different earthquakes easily. Understanding logarithms helps us make sense of phenomena that span vast scales.
To find lo g 10 16 , we express 16 as 8 3 4 and use the logarithm property to get lo g 10 16 = 3 4 lo g 10 8 . Substituting the value lo g 10 8 = 0.9031 and calculating gives us lo g 10 16 ≈ 1.2041 .
;
