[tex]$\log _{10} 8=0.9031$[/tex], Find [tex]$4+\log _{10} 16$[/tex]
Answer (2)
Express 16 as 2 4 and 8 as 2 3 .
Use the logarithm property lo g ( a b ) = b lo g ( a ) .
Find lo g 10 2 = 3 0.9031 .
Calculate 4 + lo g 10 16 = 4 + 4 ( 3 0.9031 ) = 5.20413333... , so the final answer is 5.2041 .
Explanation
Problem Analysis We are given that lo g 10 8 = 0.9031 , and we want to find the value of 4 + lo g 10 16 .
Expressing 16 as a power 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 . Therefore, we can write lo g 10 16 = lo g 10 ( 2 4 ) = 4 lo g 10 2 .
Finding log base 10 of 2 Also, we have lo g 10 8 = lo g 10 ( 2 3 ) = 3 lo g 10 2 = 0.9031 . From this, we can find lo g 10 2 = 3 0.9031 .
Calculating log base 10 of 16 Now we can substitute this value into the expression for lo g 10 16 : lo g 10 16 = 4 lo g 10 2 = 4 × 3 0.9031 = 3 4 × 0.9031 = 3 3.6124 = 1.20413333...
Calculating the final value Finally, we can find the value of the expression 4 + lo g 10 16 : 4 + lo g 10 16 = 4 + 1.20413333... = 5.20413333...
Final Answer Therefore, the value of 4 + lo g 10 16 is approximately 5.2041 .
Examples
Logarithms are incredibly useful in many real-world situations. For example, they are used to measure the magnitude of earthquakes on the Richter scale. The Richter scale is logarithmic, meaning that an increase of one unit on the scale corresponds to a tenfold increase in the amplitude of the earthquake waves. Logarithms are also used in chemistry to measure pH levels, in acoustics to measure sound intensity, and in finance to calculate compound interest. Understanding logarithms helps us to quantify and analyze phenomena that vary over a wide range of values.
To find 4 + lo g 10 16 , we express 16 as 2 4 and use given information on lo g 10 8 to find lo g 10 2 . This leads to the calculation 4 + 4 lo g 10 2 ≈ 5.2041 . Thus, the answer is 5.2041 .
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