The two expressions below have the same value when rounded to the nearest hundredth.
[tex]$\log _5 b \quad \log _9 48$[/tex]
What is the approximate value of [tex]$\log b$[/tex] to the nearest hundredth?
A. 0.93
B. 1.23
C. 9.16
D. 65.53
Answer (2)
Calculate lo g 9 48 ≈ 1.761859507 .
Use the fact that lo g 5 b ≈ 1.761859507 .
Calculate lo g b = lo g ( 5 1.761859507 ) = 1.761859507 × lo g 5 ≈ 1.761859507 × 0.698970004 ≈ 1.231486947 .
Round to the nearest hundredth: lo g b ≈ 1.23 .
Explanation
Understanding the Problem We are given that lo g 5 b and lo g 9 48 have the same value when rounded to the nearest hundredth. Our goal is to find the approximate value of lo g b to the nearest hundredth.
Calculating log_9 48 First, let's find the value of lo g 9 48 . Using a calculator, we find that lo g 9 48 ≈ 1.761859507 .
Equating the Logarithms Since lo g 5 b has the same value as lo g 9 48 when rounded to the nearest hundredth, we have lo g 5 b ≈ 1.76 . Therefore, we can say that lo g 5 b = 1.761859507 to a higher degree of accuracy.
Rewriting the Equation We want to find lo g b . We know that lo g 5 b = 1.761859507 . We can rewrite this as b = 5 1.761859507 .
Taking the Logarithm Now, we take the logarithm (base 10) of both sides: lo g b = lo g ( 5 1.761859507 ) = 1.761859507 × lo g 5 .
Calculating log b We know that lo g 5 ≈ 0.698970004 . Therefore, lo g b ≈ 1.761859507 × 0.698970004 ≈ 1.231486947 .
Rounding the Result Rounding to the nearest hundredth, we get lo g b ≈ 1.23 .
Final Answer Therefore, the approximate value of lo g b to the nearest hundredth is 1.23 .
Examples
Logarithms are incredibly useful in many real-world applications. For example, they are used in seismology to measure the magnitude of earthquakes on the Richter scale. They also appear in finance when calculating compound interest and in chemistry when determining the pH of a solution. Understanding logarithms helps us quantify and analyze phenomena that span vast ranges of values.
The approximate value of lo g b to the nearest hundredth is 1.23 .
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