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

2024
Views:

95

Pages:

3
Author:

Reynaldo Wong

Question 1 Use the properties of logarithms to simplify: log ⁡ 32 2 − log ⁡ 20 0.025 log 32 2−log 20 0.025 Answer 1 Step 1: Logarithms are inverse of exponential functions because log ⁡ 𝑎 ( 𝑎 𝑥 ) = 𝑥 log a (a x )=x. Any logarithmic equation can be written into an equivalent exponential equation. We will use: log ⁡ 𝑎 ( 𝑏 𝑐 ) = log ⁡ 𝑎 𝑏 − log ⁡ 𝑎 𝑐 log a ( c b )=log a b−log a c Also, 128 = 2 7 128=2 7 . Step 2: Expression to evaluate: log ⁡ 32 2 − log ⁡ 20 0.025 log 32 2−log 20 0.025 Simplify: log ⁡ 32 2 = 1 5 , log ⁡ 20 0.025 = − 2 log 32 2= 5 1 ,log 20 0.025=−2 Calculate: 1 5 − ( − 2 ) = 1 5 + 2 = 7 5 1 −(−2)= 5 1 +2=7 Final Answer: 7 7 Question 2 Use the change-of-base formula to write ( log ⁡ 25 log ⁡ 55 ) (log 25 log 55 ) as a single logarithm. Answer 1 Step 1: The change-of-base formula: log ⁡ 𝑏 𝑎 = ln ⁡ 𝑎 ln ⁡ 𝑏 log b a= lnb lna Step 2: Rewrite the given expression: ( log ⁡ 25 log ⁡ 55 ) = ln ⁡ ( ln ⁡ 55 ) ln ⁡ 25 (log 25 log 55 )= ln25 ln(ln55) Question 3 Use the laws of logarithms to expand the expression: log ⁡ 𝑦 3 2 𝑥 log 2x y 3 Answer 1 Step 1: Apply the properties of logarithms: log ⁡ 𝑦 3 2 𝑥 = log ⁡ ( 𝑦 3 ) − log ⁡ ( 2 𝑥 ) log 2x y 3 =log(y 3 )−log( 2x ) Step 2: Simplify: log ⁡ ( 𝑦 3 ) = 3 log ⁡ ( 𝑦 ) , log ⁡ ( 2 𝑥 ) = 1 2 log ⁡ ( 2 𝑥 ) = 1 2 [ log ⁡ ( 2 ) + log ⁡ ( 𝑥 ) ] log(y 3 )=3log(y),log( 2x )= 2 1 log(2x)= 2 1 [log(2)+log(x)] Combine terms: 3 log ⁡ ( 𝑦 ) − 1 2 [ log ⁡ ( 2 ) + log ⁡ ( 𝑥 ) ] = 3 log ⁡ ( 𝑦 ) − 1 2 log ⁡ ( 2 ) − 1 2 log ⁡ ( 𝑥 ) 3log(y)− 2 1 [log(2)+log(x)]=3log(y)− 2 1 log(2)− 2 1 log(x) Final Answer: 3 log ⁡ ( 𝑦 ) − 1 2 log ⁡ ( 2 ) − 1 2 log ⁡ ( 𝑥 ) 3log(y)− 2 1 log(2)− 2 1 log(x) Question 4 Use common logarithms or natural logarithms and a calculator to evaluate to four decimal places: log ⁡ 16 57.2 log 16 57.2 Answer 1 Step 1: Using the change-of-base formula: log ⁡ 16 ( 57.2 ) = log ⁡ ( 57.2 ) log ⁡ ( 16 ) log 16 (57.2)= log(16) log(57.2) Step 2: Using a calculator: log ⁡ ( 57.2 ) ≈ 1.7574 , log ⁡ ( 16 ) ≈ 1.2041 log(57.2)≈1.7574,log(16)≈1.2041 Divide: log ⁡ 16 ( 57.2 ) = 1.7574 1.2041 ≈ 1.4595 log 16 (57.2)= 1.2041 1.7574 ≈1.4595 Final Answer: 1.4595 1.4595 Question 5 Use properties of logarithms to expand the logarithmic expression: log ⁡ 5 ( 13 ⋅ 7 ) log 5 (13⋅7) Answer 1 Step 1: Using the product rule: log ⁡ 𝑏 ( 𝑚 ⋅ 𝑛 ) = log ⁡ 𝑏 ( 𝑚 ) + log ⁡ 𝑏 ( 𝑛 ) log b (m⋅n)=log b (m)+log b (n) Step 2: Apply the product rule: log ⁡ 5 ( 13 ⋅ 7 ) = log ⁡ 5 ( 13 ) + log ⁡ 5 ( 7 ) log 5 (13⋅7)=log 5 (13)+log 5 (7) Final Answer: log ⁡ 5 ( 13 ) + log ⁡ 5 ( 7 ) log 5 (13)+log 5 (7)