If [tex]$\log 32=1.505$[/tex], then what is the value of [tex]$\log _{1000} 32$[/tex]?
A. 0.502
B. 001505
C. .000502
D. 1,505
Answer (2)
Use the change of base formula to express lo g 1000 32 as l o g 1000 l o g 32 .
Determine that lo g 32 = 1.505 and lo g 1000 = 3 .
Substitute these values into the expression: 3 1.505 .
Calculate the final value: 3 1.505 = 0.501666... ≈ 0.502 . The value of lo g 1000 32 is 0.502 .
Explanation
Understanding the problem We are given that lo g 32 = 1.505 and we want to find the value of lo g 1000 32 .
Applying the change of base formula We can use the change of base formula to rewrite lo g 1000 32 in terms of lo g 32 . The change of base formula states that lo g a b = l o g c a l o g c b for any positive a , b , and c where a = 1 and c = 1 . In our case, we can rewrite lo g 1000 32 as l o g 1000 l o g 32 .
Finding log 1000 We are given that lo g 32 = 1.505 . We need to find lo g 1000 . Since the base of the logarithm is not specified, we assume it is base 10. Therefore, lo g 1000 = lo g 10 1000 = lo g 10 1 0 3 = 3 .
Substituting the values Now we can substitute the values we found into the expression: lo g 1000 32 = l o g 1000 l o g 32 = 3 1.505 .
Calculating the final value Now we calculate the value of the fraction: 3 1.505 = 0.501666... ≈ 0.502 .
Final Answer Therefore, the value of lo g 1000 32 is approximately 0.502 .
Examples
Logarithms are incredibly useful in many real-world situations. For example, they can be used to measure the magnitude of earthquakes on the Richter scale, which is a base-10 logarithmic scale. They are also used in chemistry to measure pH levels, and in finance to calculate compound interest. Understanding logarithms helps us to quantify and compare values across a wide range of scales.
The value of lo g 1000 32 is approximately 0.502 when calculated using the change of base formula. We find this by substituting lo g 32 = 1.505 and lo g 1000 = 3 into the formula. So, the answer is option A: 0.502.
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