University:
Stanford University
Course:
MATH 19 | Calculus
Academic year:

2023
Views:

29

Pages:

15
Author:

MathPaladin

Questions and Answers Sheet 4 Logarithms Question #1 Solve the given equation for x. ln 𝑥 2 − ln 2 𝑥 + 1 = 0 Answer: Question #2 Solve for 𝑥, log 2 (2𝑥 + 10) = 4 Answer: Question #3 Use properties of logarithms to expand the logarithmic expression log 8 (13 ⋅ 7) as much as possible. Where possible, evaluate logarithmic expressio ns without using a calculator. Answer: Step 1 Use properties of logarithms to expand the logarithmic expression log 8 (13 ⋅ 7) as much as possible. Step 2 The given logarithmic expression is log 8 (13 ⋅ 7) The product rules states that the logarithm of a product is the sum of the logarithms. log 𝑚 (𝑎 ⋅ 𝑏) = log 𝑚 (𝑎) + log 𝑚 (𝑏) Apply the product rule in the given expression. log 8 (13 ⋅ 7) = log 8 (13) + log 8 (7) The expanded form of log 8 (13 ⋅ 7) is log 8 (13) + log 8 (7) Question #4 Add notes Use the properties of logarithms to rewrite the logarithm if possible. Assume that all variables represent positive real numbers. log 6 13 9 Answer: Question #5 Use the Laws of Logarithms to combine the expression into a single logarithm. a) log 𝑎 + 2 log 𝑏 b) ln(𝑥 2 − 25) − ln(𝑥 + 5) 1 c) log 2 3 − 3 log 2 𝑥 + log 2 (𝑥 + 1) 2 Answer: Question #6 1050𝑒 0.055𝑥 = 3150 What is the solution in terms of natural logarithms? Answer: Question #7 Evaluate the expression. a) log 733 b) log 464 c) log 5125 Answer: Step 1 To evaluate the given expressions, the following properties of logarithm will be used: The exponent property logmax = m log ax and the property log aa = 1 Note that the base of the logarithm must be positive. Step 2 (a) The given expression is log 733 . Use the properties of logarithms to evaluate this expression. 7 log33 = 7log33 = 7⋅1 =7 (b) The given expression is log 464 . Use the properties of logarithms to evaluate the expression. log464 = log344 = 3log 44 = 3⋅1 =3 (c) The given expression is log 5125 . Use the properties of logarithms to evaluate the expression. 3 log5125 = log55 = 3log 55 = 3⋅1 =3 Question #8 Difference Quotient 1 If 𝑓(𝑥) = log 𝑎 𝑥, show that 𝑓(𝑥+ℎ)−𝑓(ℎ) ℎ = log 𝑎 (1 + ℎ ℎ ) ,ℎ 𝑥 ≠0 Answer: Step 1 For a function f the difference quotient 𝑓(𝑥+ℎ)−𝑓(𝑥) ℎ is used to calculate the derivative by taking the limit ⇒ 0. To compute difference quotient for given functions properties of logarithms will be used. Exponent property of logarithms is 𝑦 log 𝑥 = log 𝑥 𝑦 . Difference property of logarithms 𝑖𝑠 log 𝑥 − log 𝑦 = 𝑥 log These properties are true for logarithm of any base. 𝑦 Step 2 Given function is 𝑓(𝑥) = log 𝑎𝑥 . Use the properties of logarithm given in step 1 to prove the given statement. 𝑓(𝑥 + ℎ) − 𝑓(𝑥) 𝑙𝑜𝑔𝑎 (𝑥 + ℎ) − log 𝑥 = ℎ ℎ = 𝑙𝑜𝑔𝑎 𝑥+ℎ 𝑥 ℎ 1 ℎ log 𝑎 (1 + ) ℎ 𝑥 1 ℎ ℎ = log 𝑎 (1 + ) 𝑥 Hence, proved. Question #9 Use the Laws of Logarithms to expand the expression. log ( 𝑥7𝑦2 𝑧7 ) Answer: Question #10 Question Express in terms of logarithms of x, y, z, or w. log 3 5√𝑦 Answer: Step 1 Express in terms of logarithms of x, y, z, or w. Given expression is log 3 5√𝑦 Step 2 So, we have to express in terms of logarithms 1 we have, log 3 5√𝑦 = log 3 (𝑦)5 1 = log 3 𝑦 5 Question #11 Use the Laws of Logarithms to combine the expression. ln(𝑎 + 𝑏) + ln(𝑎 − 𝑏) − 2 ln 𝑐 Answer: Question #12 Use common logarithms or natural logarithms and a calculator to evaluate to four decimal places: log 5 1 3 Answer: Step 1 Evaluate: log 5 (13) Step 2 So, log 5 (13) = = = ln 13 ln 5 2.565 13 5 [since log 𝑏 𝑎 = log 𝑎 log 𝑏 ] [Natural logarithms] 1.609 =1.59 Step 3 Hence, log 5 (13) = 1.59 Question #13 Show that log 𝑎 (12𝑥 + 2𝑥 − 12) + log 𝑎 (12𝑥 − 2𝑥 − 12) = 0 Answer: Step 1 given: log 𝑎 (12𝑥 + 2𝑥 − 12) + log 𝑎 (12𝑥 − 2𝑥 − 12) = 0 The product rule for logarithms can be used to simplify a logarithm of a product by rewriting it as a sum of individual logarithms. log 𝑏 (𝑚𝑛) = log 𝑏 (𝑚) + log 𝑏 (𝑛) Step 2 log 𝑎 (12𝑥 + 2𝑥 − 12) + log 𝑎 (12𝑥 − 2𝑥 − 12) = 0 Use product rule Question #14 Write each expression as a single logarithms. 2 log 2 (𝑥 + 1) − log 2 (𝑥 + 3) − log 2 (𝑥 − 1) Answer: Step 1 Given Expression 2 log 2 (𝑥 + 1) − log 2 (𝑥 + 3) − log 2 (𝑥 − 1) Step 2 To Write The Given expression as a single logarithms Step 3 Formula Used 𝑎 log 𝑎 − log 𝑏 = log ( ) 𝑏 𝑚 log 𝑎 = log 𝑎𝑚 log 𝑎 + log 𝑏 = log(𝑎𝑏) Step 4 Solution to the question We have, 2 log 2 (𝑥 + 1) − log 2 (𝑥 + 3) − log 2 (𝑥 − 1) = log 2 (𝑥 + 1)2 − log 2 (𝑥 + 3) − log 2 (𝑥 − 1) = log 2 (𝑥+1)2 − log 2 (𝑥 − 1) 𝑥+3 (𝑥+1)2 = log 2 (𝑥+3)(𝑥−1) Question #15 Solve the following exponential equation. Express the solution set in terms of natural logarithms or common logarithms. Then, use a calculator to obtain a decimal approximation for the solution. 70.7𝑥 = 597 The solution set expressed in terms of logarithms is { }. (Use a comma to separate answers as needed. Simplify your answer. Use integers or decimals for any numbers in the expression. Use In for natural logarithm and log for common logarithm.) Answer: Step 1 Given, 70.7𝑥 = 597 Step 2 Taking natural logarithm on both side, ln(70.7𝑥 ) = ln(597) 0.7𝑥 ln(7) = ln(597) 0.7𝑥 = 𝑥= 𝑥= 𝑥= 𝑥= ln(597) ln(7) ln(597) ln(7)⋅(0.7) ln(597) ln(7)⋅( 7 ) 10 ln(597)⋅10 7 ln(7) 10 ln(597) 7 ln(7) Step 3 Using calculator, ln(597)⋅10 7 ln(7) ≈ 4.693 Step 4 Therefore, the solution set in term of logarithms is 10 ln(597) 7 ln(7) Question #16 Write the equation in its equivalent exponential form. 4 = log 5 𝑀 Answer: Step 1 It is logarithms. Step 2 Given: 4 = log 5 𝑀 4 = log 5 𝑀 54 = 𝑀 𝑀 = 54 Note: log 𝑎 𝑏 = 𝑐 ⇒ 𝑎𝑐 = 𝑏 Question #17 Find the logarithm. log 6 2 16 Answer: Concept Properties of logarithms: 1. log(𝑎)𝑏 = 𝑏 log(𝑎) 2. log 𝑎 (𝑎) = 1 Solution To evaluate: log 6 216 = log 6 (6)3 = 3 log 6 (6) {∵ log(𝑎)𝑏 = 𝑏 log(𝑎) here 𝑎 = 6, 𝑏 = 3} = 3×1 = 3 {∵ log 𝑎 (𝑎) = 1} Hence, log 6 216 = 3 Question #18 Use the Laws of Logarithms to evaluate the expression. log 3 (15) − log 3 (40) + log 3 (216) Answer: Step 1 using the law of logarithms: Step 2 log 3 (15) − log 3 (40) + log 3 (216) 𝑎 Applying log rule: log 𝑐 (𝑎) − log 𝑐 (𝑏) = log 𝑐 ( ) 𝑏 15 = log 3 ( ) + log 3 (216) 40 3 = log 3 ( ) + log 3 (216) 8 Applying log rule: log 𝑐 (𝑎) + log 𝑐 (𝑏) = log 𝑐 (𝑎𝑏) 3 = log 3 (216 ⋅ ) 8 = 𝑙𝑜𝑔3 (81) = log 3 (3)4 Applying log rule: log 𝑎 (𝑥)𝑏 = 𝑏 ⋅ log 𝑎 (𝑥) = 4 log 3 (3) Applying log rule: log 𝑎 (𝑎) = 1 = 4 ⋅ log 3 (3) =4 Question #19 To clear the following equation of fractions, by what should both sides be multiplied? 7 2𝑦−38 +𝑦= 10 5𝑦−25 Answer: Question #20 If log 𝑎 (𝑥) = 1.5 and log 𝑎 (𝑦) = 4.7 find the following using the properties of logarithms. 𝑥 log 𝑎 ( ) 𝑦 Answer: Step 1 properties of logarithms 𝑥 log𝑎 (𝑥) 𝑦 log𝑎 (𝑦) log 𝑎 ( ) = Now log 𝑎 (𝑥) = 1.5 log 𝑎 (𝑦) = 4.7 Step 2 from property 𝑥 log 𝑎 ( ) = = 𝑦 log𝑎 (𝑥) log𝑎 (𝑦) 1.5 4.7 = 0.3191 𝑥 log 𝑎 ( ) = 0.3191 Answer 𝑦