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

2024
Views:

177

Pages:

4
Author:

Reynaldo Wong

Question 1 What is the difference between log and ln? Answer 1 Log ( log ⁡ 10 log 10 ): Usually log ⁡ ( 𝑥 ) log(x) refers to the base-10 logarithm (common logarithm). It tells you what power you must raise 10 to in order to obtain the number 𝑥 x. Example: 1 0 𝑦 = 𝑥 ⟹ log ⁡ 10 ( 𝑥 ) = 𝑦 10 y =x⟹log 10 (x)=y Ln ( log ⁡ 𝑒 log e ): The natural logarithm, denoted ln ⁡ ( 𝑥 ) ln(x), is the logarithm with base 𝑒 ≈ 2.718 e≈2.718. It tells you what power you must raise 𝑒 e to in order to obtain 𝑥 x. Example: 𝑒 𝑦 = 𝑥 ⟹ ln ⁡ ( 𝑥 ) = 𝑦 e y =x⟹ln(x)=y Question 2 How do I find the common logarithm of a number? Answer 1 To find the common logarithm (log base 10): log ⁡ 10 ( 𝑥 ) = ln ⁡ ( 𝑥 ) ln ⁡ ( 10 ) log 10 (x)= ln(10) ln(x) If you have a calculator with ln ⁡ ( 𝑥 ) ln(x), use the formula above. Alternatively, use the following approximation for ln ⁡ ( 10 ) ln(10): ln ⁡ ( 10 ) ≈ 2.3026 ln(10)≈2.3026 Example: For 𝑥 = 1000 x=1000: log ⁡ 10 ( 1000 ) = 3 (since 1 0 3 = 1000 ) log 10 (1000)=3(since 10 3 =1000) Question 3 Calculating how long it will take for the half-life of 𝑥 x amount to fall below 𝑦 y: Setup: You are trying to determine the time ( 𝑡 t) for 500 units of substance to fall below 100. 99 = 500 ( 1 2 ) 𝑡 99=500( 2 1 ) t Answer 1 The half-life equation is: 𝐴 = 𝐴 0 ⋅ ( 1 2 ) 𝑡 A=A 0 ⋅( 2 1 ) t In this case: 99 = 500 ⋅ ( 1 2 ) 𝑡 ⟹ ( 1 2 ) 𝑡 = 99 500 99=500⋅( 2 1 ) t ⟹( 2 1 ) t = 500 99 Take the natural logarithm: ln ⁡ ( ( 1 2 ) 𝑡 ) = ln ⁡ ( 99 500 ) ln(( 2 1 ) t )=ln( 500 99 ) Simplify: 𝑡 ⋅ ln ⁡ ( 1 2 ) = ln ⁡ ( 99 500 ) t⋅ln( 2 1 )=ln( 500 99 ) 𝑡 = ln ⁡ ( 99 500 ) ln ⁡ ( 1 2 ) t= ln( 2 1 ) ln( 500 99 ) Substitute ln ⁡ ( 1 2 ) = − ln ⁡ ( 2 ) ln( 2 1 )=−ln(2): 𝑡 = ln ⁡ ( 99 ) − ln ⁡ ( 500 ) − ln ⁡ ( 2 ) t= −ln(2) ln(99)−ln(500) Question 4 Logarithm rules: Which has priority? Compare 2 𝑥 ( ln ⁡ 2 + ln ⁡ 𝑒 ) 2x(ln2+lne) and 2 𝑥 ( ln ⁡ 2 𝑒 ) 2x(ln2e). Answer 1 The correct interpretation is: 2 𝑥 ( ln ⁡ 2 + ln ⁡ 𝑒 ) = 2 𝑥 ln ⁡ ( 2 𝑒 ) 2x(ln2+lne)=2xln(2e) The logarithmic property ln ⁡ ( 𝑎 ) + ln ⁡ ( 𝑏 ) = ln ⁡ ( 𝑎 𝑏 ) ln(a)+ln(b)=ln(ab) allows us to combine the terms. Question 5 How to solve log ⁡ 50 + log ⁡ ( 𝑥 2 ) = 0 log50+log( 2 x )=0? Answer 1 log ⁡ ( 50 ) + log ⁡ ( 𝑥 2 ) = 0 log(50)+log( 2 x )=0 Using the product rule: log ⁡ ( 50 ⋅ 𝑥 2 ) = 0 log(50⋅ 2 x )=0 Simplify: log ⁡ ( 50 𝑥 2 ) = log ⁡ ( 1 ) log( 2 50x )=log(1) 50 𝑥 2 = 1 ⟹ 25 𝑥 = 1 ⟹ 𝑥 = 1 25 2 50x =1⟹25x=1⟹x= 25 1 Final Answer: 𝑥 = 0.04 x=0.04