(b) [tex]log _5(30)=[/tex]

Express 30 as a product of its prime factors.
Apply logarithm product rule.
Use change of base formula.
Calculate the final answer: 2.11328 ​

Explanation

Understanding the problem We are asked to evaluate the logarithm of 30 with base 5, which is written as lo g 5 ​ ( 30 ) .

Using logarithm properties We can express 30 as a product of its prime factors: 30 = 2 ⋅ 3 ⋅ 5 . Using the properties of logarithms, we can rewrite the expression as: lo g 5 ​ ( 30 ) = lo g 5 ​ ( 2 ⋅ 3 ⋅ 5 ) = lo g 5 ​ ( 2 ) + lo g 5 ​ ( 3 ) + lo g 5 ​ ( 5 ) Since lo g 5 ​ ( 5 ) = 1 , we have lo g 5 ​ ( 30 ) = lo g 5 ​ ( 2 ) + lo g 5 ​ ( 3 ) + 1

Applying change of base formula Alternatively, we can use the change of base formula to express the logarithm in terms of natural logarithms (ln) or common logarithms (log base 10). The change of base formula is: lo g b ​ ( a ) = lo g c ​ ( b ) lo g c ​ ( a ) ​ In our case, we can use natural logarithms: lo g 5 ​ ( 30 ) = ln ( 5 ) ln ( 30 ) ​

Calculating the value Using a calculator, we find that lo g 5 ​ ( 30 ) = ln ( 5 ) ln ( 30 ) ​ ≈ 1.609438 3.401197 ​ ≈ 2.11328

Final Answer Therefore, the value of lo g 5 ​ ( 30 ) is approximately 2.11328.


Examples
Logarithms are used in many real-world applications, such as measuring the intensity of earthquakes (Richter scale), the loudness of sound (decibels), and the acidity of a solution (pH scale). They are also used in computer science to analyze the efficiency of algorithms and in finance to calculate compound interest. For example, if you want to determine how long it will take for an investment to double at a certain interest rate, you can use logarithms to solve for the time. The formula for compound interest is A = P ( 1 + r ) t , where A is the final amount, P is the principal, r is the interest rate, and t is the time. Solving for t involves using logarithms.

Answered by GinnyAnswer | 2025-07-04

To find lo g 5 ​ ( 30 ) , we can express 30 as a product of its prime factors and use logarithmic properties. By applying the change of base formula, we calculate it as approximately 2.11328. This involves both prime factorization and logarithmic calculations using natural logarithms.
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Answered by Anonymous | 2025-07-06