University:
Morehouse College
Course:
HMTH 105 | College Algebra 1
Academic year:

2024
Views:

314

Pages:

3
Author:

Reynaldo Wong

Question 1 Write the expression as a single logarithm: 6 log ⁡ 5 ( 𝑥 − 3 ) − 4 log ⁡ 5 ( 𝑥 + 4 ) 6log 5 (x−3)−4log 5 (x+4) Answer 1 Step 1: Express the given quantity as the logarithm of a single expression. Step 2: Apply logarithmic rules: 𝑚 log ⁡ 𝑎 𝑝 = log ⁡ 𝑎 ( 𝑝 𝑚 ) , log ⁡ 𝑎 𝑝 − log ⁡ 𝑎 𝑞 = log ⁡ 𝑎 ( 𝑝 𝑞 ) mlog a p=log a (p m ),log a p−log a q=log a ( q p ) Step 3: Simplify: 6 log ⁡ 5 ( 𝑥 − 3 ) = log ⁡ 5 ( ( 𝑥 − 3 ) 6 ) , 4 log ⁡ 5 ( 𝑥 + 4 ) = log ⁡ 5 ( ( 𝑥 + 4 ) 4 ) 6log 5 (x−3)=log 5 ((x−3) 6 ),4log 5 (x+4)=log 5 ((x+4) 4 ) Combine: log ⁡ 5 ( ( 𝑥 − 3 ) 6 ) − log ⁡ 5 ( ( 𝑥 + 4 ) 4 ) = log ⁡ 5 ( ( 𝑥 − 3 ) 6 ( 𝑥 + 4 ) 4 ) log 5 ((x−3) 6 )−log 5 ((x+4) 4 )=log 5 ( (x+4) 4 (x−3) 6 ) Final Answer: log ⁡ 5 ( ( 𝑥 − 3 ) 6 ( 𝑥 + 4 ) 4 ) log 5 ( (x+4) 4 (x−3) 6 ) Question 2 State the product property of logarithms. Answer 1 The product property of logarithms states: log ⁡ 𝑏 ( 𝑥 𝑦 ) = log ⁡ 𝑏 ( 𝑥 ) + log ⁡ 𝑏 ( 𝑦 ) , for positive real numbers 𝑥 , 𝑦 and 𝑏 > 1. log b (xy)=log b (x)+log b (y),for positive real numbers x,y and b>1. Question 3 Use properties of logarithms to expand the logarithmic expression as much as possible: log ⁡ 5 ( 625 𝑦 ) log 5 ( y 625 ) Answer 1 Step 1: Apply the quotient rule: log ⁡ 𝑏 ( 𝑎 𝑐 ) = log ⁡ 𝑏 ( 𝑎 ) − log ⁡ 𝑏 ( 𝑐 ) log b ( c a )=log b (a)−log b (c) log ⁡ 5 ( 625 𝑦 ) = log ⁡ 5 ( 625 ) − log ⁡ 5 ( 𝑦 ) log 5 ( y 625 )=log 5 (625)−log 5 (y) Step 2: Evaluate log ⁡ 5 ( 625 ) log 5 (625): 625 = 5 4 ⟹ log ⁡ 5 ( 625 ) = 4 625=5 4 ⟹log 5 (625)=4 Final Answer: 4 − log ⁡ 5 ( 𝑦 ) 4−log 5 (y) Question 4 Use the properties of logarithms to condense the logarithmic expression: log ⁡ 3 ( 𝑥 2 − 1 ) − 2 log ⁡ 3 ( 𝑥 − 1 ) log 3 (x 2 −1)−2log 3 (x−1) Answer 1 Step 1: Use the power rule: 2 log ⁡ 3 ( 𝑥 − 1 ) = log ⁡ 3 ( ( 𝑥 − 1 ) 2 ) 2log 3 (x−1)=log 3 ((x−1) 2 ) Step 2: Apply the quotient rule: log ⁡ 3 ( 𝑥 2 − 1 ) − log ⁡ 3 ( ( 𝑥 − 1 ) 2 ) = log ⁡ 3 ( 𝑥 2 − 1 ( 𝑥 − 1 ) 2 ) log 3 (x 2 −1)−log 3 ((x−1) 2 )=log 3 ( (x−1) 2 x 2 −1 ) Step 3: Simplify the fraction: 𝑥 2 − 1 = ( 𝑥 − 1 ) ( 𝑥 + 1 ) x 2 −1=(x−1)(x+1) 𝑥 2 − 1 ( 𝑥 − 1 ) 2 = ( 𝑥 − 1 ) ( 𝑥 + 1 ) ( 𝑥 − 1 ) 2 = 𝑥 + 1 𝑥 − 1 (x−1) 2 x 2 −1 = (x−1) 2 (x−1)(x+1) = x−1 x+1 Final Answer: log ⁡ 3 ( 𝑥 + 1 𝑥 − 1 ) log 3 ( x−1 x+1 ) Question 5 Rewrite the expression as a single logarithm and simplify the result: ln ⁡ ∣ sin ⁡ 𝑥 ∣ + ln ⁡ ∣ cot ⁡ 𝑥 ∣ ln∣sinx∣+ln∣cotx∣ Answer 1 Step 1: Apply the product rule: ln ⁡ ∣ sin ⁡ 𝑥 ∣ + ln ⁡ ∣ cot ⁡ 𝑥 ∣ = ln ⁡ ( ∣ sin ⁡ 𝑥 ∣ ⋅ ∣ cot ⁡ 𝑥 ∣ ) ln∣sinx∣+ln∣cotx∣=ln(∣sinx∣⋅∣cotx∣) Step 2: Simplify ∣ cot ⁡ 𝑥 ∣ = ∣ cos ⁡ 𝑥 ∣ ∣ sin ⁡ 𝑥 ∣ ∣cotx∣= ∣sinx∣ ∣cosx∣ : ln ⁡ ( ∣ sin ⁡ 𝑥 ∣ ⋅ ∣ cot ⁡ 𝑥 ∣ ) = ln ⁡ ( ∣ sin ⁡ 𝑥 ∣ ⋅ ∣ cos ⁡ 𝑥 ∣ ∣ sin ⁡ 𝑥 ∣ ) ln(∣sinx∣⋅∣cotx∣)=ln( ∣sinx∣ ∣sinx∣⋅∣cosx∣ ) Cancel ∣ sin ⁡ 𝑥 ∣ ∣sinx∣: ln ⁡ ∣ cos ⁡ 𝑥 ∣ ln∣cosx∣ Final Answer: ln ⁡ ∣ cos ⁡ 𝑥 ∣ ln∣cosx∣