Given that [tex]$\log _b(8) \approx 2.079, \log _b(9) \approx 2.197$[/tex], and [tex]$\log _b(17) \approx 2.833$[/tex], find [tex]$\log _b \frac{1}{72}$[/tex].
[tex]$\log _b \frac{1}{72} \approx$[/tex]
(Simplify your answer. Round to the nearest thousandth.)
Answer (2)
Express 72 as a product of 8 and 9 , so 72 1 = 8 × 9 1 .
Apply logarithm properties to rewrite the expression: lo g b 72 1 = − ( lo g b 8 + lo g b 9 ) .
Substitute the given values: lo g b 8 ≈ 2.079 and lo g b 9 ≈ 2.197 .
Calculate the final result: lo g b 72 1 ≈ − ( 2.079 + 2.197 ) = − 4.276 .
Explanation
Problem Setup We are given the approximations lo g b ( 8 ) ≈ 2.079 and lo g b ( 9 ) ≈ 2.197 , and we want to find lo g b 72 1 .
Expressing 1/72 as a product First, we can express 72 as a product of 8 and 9: 72 = 8 × 9 . Therefore, 72 1 = 8 × 9 1 .
Applying Logarithm Properties Now, we use the logarithm property that lo g b ( x y ) = lo g b ( x ) + lo g b ( y ) and lo g b ( x 1 ) = − lo g b ( x ) . Thus, we have lo g b 72 1 = lo g b 8 × 9 1 = − lo g b ( 8 × 9 ) = − ( lo g b 8 + lo g b 9 )
Substituting Given Values We are given that lo g b ( 8 ) ≈ 2.079 and lo g b ( 9 ) ≈ 2.197 . Substituting these values, we get lo g b 72 1 ≈ − ( 2.079 + 2.197 ) = − 4.276
Final Answer Therefore, lo g b 72 1 ≈ − 4.276 .
Examples
Logarithms are incredibly useful in various fields, such as calculating the magnitude of earthquakes on the Richter scale, determining the acidity or alkalinity of a solution using pH values, and modeling exponential growth or decay in populations or investments. For instance, if you want to compare the intensity of two earthquakes, you can use the logarithm of their amplitudes to find the difference in their magnitudes. Similarly, in finance, logarithms help in calculating the time it takes for an investment to double at a certain interest rate.
To find lo g b 72 1 , we express 72 as 8 × 9 and apply logarithmic properties to get − ( lo g b 8 + lo g b 9 ) . By substituting the given approximations for logarithms, we find that lo g b 72 1 ≈ − 4.276 .
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