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

2024
Views:

59

Pages:

5
Author:

Reynaldo Wong

Question 1 Use the laws of logarithms to combine the expression into a single logarithm. a) log ⁡ 𝑎 + 2 log ⁡ 𝑏 loga+2logb b) ln ⁡ ( 𝑥 2 − 25 ) − ln ⁡ ( 𝑥 + 5 ) ln(x 2 −25)−ln(x+5) c) log ⁡ 25 3 − 3 log ⁡ 25 𝑥 + 1 2 log ⁡ 25 2 ( 𝑥 + 1 ) log 25 3−3log 25 x+ 2 1 log 25 2(x+1) Answer 1 Part a) Using the power and product rules: log ⁡ 𝑎 + 2 log ⁡ 𝑏 = log ⁡ 𝑎 + log ⁡ 𝑏 2 = log ⁡ ( 𝑎 𝑏 2 ) loga+2logb=loga+logb 2 =log(ab 2 ) Part b) Using the quotient rule: ln ⁡ ( 𝑥 2 − 25 ) − ln ⁡ ( 𝑥 + 5 ) = ln ⁡ ( 𝑥 2 − 25 𝑥 + 5 ) ln(x 2 −25)−ln(x+5)=ln( x+5 x 2 −25 ) Factor 𝑥 2 − 25 x 2 −25 as ( 𝑥 − 5 ) ( 𝑥 + 5 ) (x−5)(x+5): ln ⁡ ( ( 𝑥 − 5 ) ( 𝑥 + 5 ) 𝑥 + 5 ) = ln ⁡ ( 𝑥 − 5 ) ln( x+5 (x−5)(x+5) )=ln(x−5) Part c) Combine using the logarithmic rules: log ⁡ 25 3 − 3 log ⁡ 25 𝑥 + 1 2 log ⁡ 25 ( 2 ( 𝑥 + 1 ) ) log 25 3−3log 25 x+ 2 1 log 25 (2(x+1)) = log ⁡ 25 ( 3 ⋅ 2 ( 𝑥 + 1 ) 𝑥 3 ) =log 25 ( x 3 3⋅ 2(x+1) ) Final Combined Form for c: log ⁡ 25 ( 3 2 ( 𝑥 + 1 ) 𝑥 3 ) log 25 ( x 3 3 2(x+1) ) Question 2 Describe the power rule for logarithms and give an example. Answer 1 Step 1: The power rule for logarithms states: log ⁡ 𝑎 ( 𝑥 𝑛 ) = 𝑛 log ⁡ 𝑎 ( 𝑥 ) log a (x n )=nlog a (x) Proof: log ⁡ 𝑎 ( 𝑥 𝑛 ) = log ⁡ 𝑎 ( 𝑥 ⋅ 𝑥 ⋅ ⋯ ⋅ 𝑥 ) ( 𝑛 times ) log a (x n )=log a (x⋅x⋅⋯⋅x)(n times) Using the product rule: log ⁡ 𝑎 ( 𝑥 𝑛 ) = log ⁡ 𝑎 ( 𝑥 ) + log ⁡ 𝑎 ( 𝑥 ) + ⋯ + log ⁡ 𝑎 ( 𝑥 ) ( 𝑛 times ) log a (x n )=log a (x)+log a (x)+⋯+log a (x)(n times) = 𝑛 log ⁡ 𝑎 ( 𝑥 ) =nlog a (x) Example: log ⁡ 10 ( 6 5 ) = 5 log ⁡ 10 ( 6 ) log 10 (6 5 )=5log 10 (6) log ⁡ 3 ( 3 4 ) = 4 log ⁡ 3 ( 3 ) = 4 log 3 (3 4 )=4log 3 (3)=4 Question 3 Prove the power property of logarithms: log ⁡ 𝑎 ( 𝑥 𝑝 ) = 𝑝 log ⁡ 𝑎 ( 𝑥 ) log a (x p )=plog a (x) Answer 1 Let 𝑞 = log ⁡ 𝑎 ( 𝑥 ) q=log a (x). Step 1: Rewrite in exponential form: 𝑥 = 𝑎 𝑞 x=a q Raise both sides to the 𝑝 p-th power: 𝑥 𝑝 = ( 𝑎 𝑞 ) 𝑝 x p =(a q ) p Simplify using the properties of exponents: 𝑥 𝑝 = 𝑎 𝑝 𝑞 x p =a pq Take the logarithm of both sides: log ⁡ 𝑎 ( 𝑥 𝑝 ) = 𝑝 𝑞 log a (x p )=pq Substitute 𝑞 = log ⁡ 𝑎 ( 𝑥 ) q=log a (x): log ⁡ 𝑎 ( 𝑥 𝑝 ) = 𝑝 log ⁡ 𝑎 ( 𝑥 ) log a (x p )=plog a (x) Final Proof Complete. Question 4 Use properties of logarithms to condense the logarithmic expression. Write the expression as a single logarithm whose coefficient is 1. Where possible, evaluate the logarithmic expression: 8 ln ⁡ 𝑥 − 1 3 ln ⁡ 𝑦 8lnx− 3 1 lny Answer 1 Step 1: Use the power rule of logarithms: 𝑎 ln ⁡ ( 𝑏 ) = ln ⁡ ( 𝑏 𝑎 ) aln(b)=ln(b a ) 8 ln ⁡ ( 𝑥 ) = ln ⁡ ( 𝑥 8 ) , 1 3 ln ⁡ ( 𝑦 ) = ln ⁡ ( 𝑦 1 / 3 ) 8ln(x)=ln(x 8 ), 3 1 ln(y)=ln(y 1/3 ) Step 2: Use the quotient rule: ln ⁡ ( 𝑥 8 ) − ln ⁡ ( 𝑦 1 / 3 ) = ln ⁡ ( 𝑥 8 𝑦 1 / 3 ) ln(x 8 )−ln(y 1/3 )=ln( y 1/3 x 8 ) Final Answer: ln ⁡ ( 𝑥 8 𝑦 1 / 3 ) ln( y 1/3 x 8 ) Question 5 Evaluate or simplify the expression without using a calculator: 1 0 log ⁡ 10 ( 𝑥 ) 10 log 10 ( x ) Answer 1 Using the power property: 1 0 log ⁡ 10 ( 𝑥 ) = 1 0 1 2 log ⁡ 10 ( 𝑥 ) 10 log 10 ( x ) =10 2 1 log 10 (x) Simplify further: = 𝑥 1 / 2 = 𝑥 =x 1/2 = x Final Answer: 𝑥 x