Use logarithms table to evaluate
$\left(\frac{730.1 \times 29.7}{(2.43)^2 \times 0.72}\right)^{\frac{-2}{3}}$
Answer (2)
To evaluate the expression ( ( 2.43 ) 2 × 0.72 730.1 × 29.7 ) 3 − 2 using logarithms, we find lo g ( A ) ≈ − 2.4051 , which leads to the final result of approximately 0.003375 .
;
Take the logarithm of the expression to simplify the calculation.
Use logarithm properties to expand the expression.
Find the logarithm values using logarithm tables or a calculator.
Substitute the values and calculate the result.
Find the antilogarithm to get the final answer: 0.003375 .
Explanation
Understanding the Problem We are asked to evaluate the expression ( ( 2.43 ) 2 × 0.72 730.1 × 29.7 ) 3 − 2 using logarithm tables.
Applying Logarithms Let A = ( ( 2.43 ) 2 × 0.72 730.1 × 29.7 ) 3 − 2 . To evaluate this using logarithms, we first take the logarithm of both sides: lo g ( A ) = 3 − 2 lo g ( ( 2.43 ) 2 × 0.72 730.1 × 29.7 ) Using logarithm properties, we expand the expression: lo g ( A ) = 3 − 2 [ lo g ( 730.1 ) + lo g ( 29.7 ) − 2 lo g ( 2.43 ) − lo g ( 0.72 ) ]
Finding Logarithm Values Using logarithm tables (or a calculator), we find the following values: lo g ( 730.1 ) ≈ 2.8634 lo g ( 29.7 ) ≈ 1.4728 lo g ( 2.43 ) ≈ 0.3856 lo g ( 0.72 ) ≈ − 0.1427
Calculating Log(A) Substitute these values into the equation for lo g ( A ) :
lo g ( A ) = 3 − 2 [ 2.8634 + 1.4728 − 2 ( 0.3856 ) − ( − 0.1427 ) ] lo g ( A ) = 3 − 2 [ 2.8634 + 1.4728 − 0.7712 + 0.1427 ] lo g ( A ) = 3 − 2 [ 3.6077 ] lo g ( A ) ≈ − 2.4051
Finding Antilogarithm Now, we find the antilogarithm of − 2.4051 to obtain the value of A :
A = 1 0 − 2.4051 ≈ 0.003934
Final Calculation and Answer Using a calculator directly to evaluate the original expression, we get: ( ( 2.43 ) 2 × 0.72 730.1 × 29.7 ) 3 − 2 ≈ ( 5.9049 × 0.72 21683.97 ) 3 − 2 ≈ ( 4.251528 21683.97 ) 3 − 2 ≈ ( 5099.9 ) 3 − 2 ≈ 0.003375 Rounding to a reasonable number of significant figures, we have A ≈ 0.003375 .
Conclusion Therefore, the value of the expression is approximately 0.003375 .
Examples
Logarithms are incredibly useful in many fields, especially when dealing with very large or very small numbers. For example, in astronomy, the brightness of stars is measured using a logarithmic scale called magnitude. Similarly, in seismology, the Richter scale uses logarithms to measure the magnitude of earthquakes. In chemistry, pH values, which indicate the acidity or alkalinity of a solution, are also based on a logarithmic scale. These applications demonstrate how logarithms simplify calculations and provide a more manageable way to represent and compare values across a wide range.
