Given that [tex]$\log _b(6) \approx 1792, \log _b(9) \approx 2.197$[/tex], and [tex]$\log _b(15) \approx 2.708$[/tex], find [tex]$\log _b \frac{1}{54}$[/tex]
[tex]$\log _b \frac{1}{54} \approx$[/tex]
(Simplify your answer. Round to the nearest thousandth.)
Answer (2)
Rewrite lo g b 54 1 as − lo g b 54 .
Express 54 as 2 ⋅ 3 3 , so − lo g b 54 = − ( lo g b 2 + 3 lo g b 3 ) .
Use lo g b 6 = lo g b 2 + lo g b 3 ≈ 1.792 and lo g b 9 = 2 lo g b 3 ≈ 2.197 to find lo g b 3 ≈ 1.0985 and lo g b 2 ≈ 0.6935 .
Substitute these values to find lo g b 54 1 ≈ − ( 0.6935 + 3 ( 1.0985 )) = − 3.989 , so the final answer is − 3.989 .
Explanation
Rewrite the expression We are given the values of lo g b ( 6 ) , lo g b ( 9 ) , and lo g b ( 15 ) , and we want to find the value of lo g b 54 1 . First, we can rewrite lo g b 54 1 as − lo g b 54 .
Prime factorization and logarithm properties Next, we express 54 as a product of its prime factors: 54 = 2 ⋅ 3 3 . Therefore, we can rewrite − lo g b 54 as − lo g b ( 2 ⋅ 3 3 ) . Using logarithm properties, this becomes − ( lo g b 2 + lo g b 3 3 ) = − ( lo g b 2 + 3 lo g b 3 ) .
Solve for log_b(3) We are given that lo g b 6 = lo g b ( 2 ⋅ 3 ) = lo g b 2 + lo g b 3 ≈ 1.792 and lo g b 9 = lo g b ( 3 2 ) = 2 lo g b 3 ≈ 2.197 . From the second equation, we can solve for lo g b 3 : lo g b 3 ≈ 2 2.197 = 1.0985
Solve for log_b(2) Now we can substitute the value of lo g b 3 into the first equation to find lo g b 2 : lo g b 2 + lo g b 3 ≈ 1.792 $\log _b 2 \approx 1.792 - 1.0985 = 0.6935 5. Calculate the final value Now we can substitute the values of $\log _b 2$ and $\log _b 3$ into the expression $-(\log _b 2 + 3 \log _b 3)$: \log _b \frac{1}{54} = -(\log _b 2 + 3 \log _b 3) \approx -(0.6935 + 3(1.0985)) = -(0.6935 + 3.2955) = -3.989
Final Answer Rounding the result to the nearest thousandth, we get -3.989.
Examples
Logarithms are incredibly useful in many real-world scenarios, such as calculating the magnitude of earthquakes using the Richter scale, determining the pH levels of chemical solutions, and modeling population growth. In finance, logarithms help in calculating continuously compounded interest rates and analyzing investment returns. Understanding logarithmic properties allows us to simplify complex calculations and make informed decisions in various fields.
We found that lo g b 54 1 ≈ − 3.989 by rewriting it as − lo g b 54 , factoring 54, and using given logarithm values to find lo g b 2 and lo g b 3 . The calculation leads to a final result of -3.989. This was achieved through properties of logarithms and calculations using provided approximations.
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