If [tex]log _{10} 8=0.9031[/tex], find...

Express 5 as a fraction: 5 = 2 10 ​ .
Use logarithm properties to rewrite the expression: lo g 10 ​ 5 = lo g 10 ​ 10 − lo g 10 ​ 2 = 1 − lo g 10 ​ 2 .
Use the given information to find lo g 10 ​ 2 : lo g 10 ​ 8 = 3 lo g 10 ​ 2 = 0.9031 , so lo g 10 ​ 2 = 0.3010333333333333 .
Substitute the value of lo g 10 ​ 2 to find lo g 10 ​ 5 : lo g 10 ​ 5 = 1 − 0.3010333333333333 = 0.6989666666666667 . Therefore, 0.6990 ​ .

Explanation

Problem Setup We are given that lo g 10 ​ 8 = 0.9031 , and we want to find lo g 10 ​ 5 .

Expressing 5 as a Fraction We know that 5 = 2 10 ​ , so we can write lo g 10 ​ 5 = lo g 10 ​ 2 10 ​ .

Applying Logarithm Properties Using the logarithm property lo g 10 ​ b a ​ = lo g 10 ​ a − lo g 10 ​ b , we have lo g 10 ​ 5 = lo g 10 ​ 10 − lo g 10 ​ 2 .

Simplifying the Expression Since lo g 10 ​ 10 = 1 , we get lo g 10 ​ 5 = 1 − lo g 10 ​ 2 . Now we need to find lo g 10 ​ 2 .

Using the Given Information We are given lo g 10 ​ 8 = 0.9031 . We can write 8 as 2 3 , so lo g 10 ​ 8 = lo g 10 ​ 2 3 .

Applying Another Logarithm Property Using the logarithm property lo g 10 ​ a b = b lo g 10 ​ a , we have lo g 10 ​ 2 3 = 3 lo g 10 ​ 2 . Therefore, 3 lo g 10 ​ 2 = 0.9031 .

Calculating log base 10 of 2 Dividing both sides by 3, we get lo g 10 ​ 2 = 3 0.9031 ​ = 0.3010333333333333 .

Finding log base 10 of 5 Now we can substitute this value back into the equation lo g 10 ​ 5 = 1 − lo g 10 ​ 2 . So, lo g 10 ​ 5 = 1 − 0.3010333333333333 = 0.6989666666666667 .

Final Answer Therefore, lo g 10 ​ 5 ≈ 0.6990 .


Examples
Logarithms are incredibly useful in many real-world applications. For instance, they are used in seismology 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. Similarly, logarithms are used in chemistry to measure pH levels, in acoustics to measure sound intensity in decibels, and in finance to calculate compound interest. Understanding logarithms helps us quantify and compare vastly different scales in a meaningful way.

Answered by GinnyAnswer | 2025-07-03

To find lo g 10 ​ 5 when lo g 10 ​ 8 = 0.9031 , we first express 5 as 2 10 ​ . By using logarithmic properties, we find lo g 10 ​ 5 = 1 − lo g 10 ​ 2 and compute lo g 10 ​ 2 based on lo g 10 ​ 8 , leading to lo g 10 ​ 5 ≈ 0.6990 .
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Answered by Anonymous | 2025-07-04