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Explanation & Answer
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1. Using the properties of exponents to prove the Power Property of Logarithms:
Solution:
Power property: log 𝑏 (𝑀𝑝 ) = p × log 𝑏 (𝑀)
Proof:
1- Let: 𝑀 = 𝑏 𝑥 which gives that log 𝑏 (𝑀) = 𝑥
2- log 𝑏 (𝑀𝑝 ) = log 𝑏 ((𝑏 𝑥 )𝑝 ) (Substitution)
= log 𝑏 (𝑏 𝑥𝑝 ) (Properties of exponents)
=𝑥𝑝
( log 𝑏 (𝑏 𝑐 ) = 𝑐)
= log 𝑏 (𝑀) × 𝑝 (Substitution)
=p × log 𝑏 (𝑀 )
Another proof:
1- Let log 𝑏 (𝑀𝑝 ) = 𝑥, then 𝑏 𝑥 = 𝑀𝑝 (Equivalent exponential form)
𝑥
𝑝
𝑝
𝑝
Then 𝑏 = 𝑀 = 𝑀 (Property of exponents)
𝑥
Then log 𝑏 (𝑀) = (Equivalent log form)
𝑝
Then p × log 𝑏 (𝑀) = 𝑥 (Multiplication by p)
Then p × log 𝑏 (𝑀) = log 𝑏 (𝑀𝑝 ) (Substitution)
2. Example of a quantity that grows or decays at a fixed rate:
Money in a bank
Investment when interest rate is fixed 𝐴 = 𝑃 × (1 + 𝑟)𝑡
where: A is the final amount, P: is the principal, r: is the interest rate, t: is the
number of years.
Real-world problem:
If...