Questions and Answers #5 Logarithms

Questions and Answers Sheet 5 Logarithms Question #1 Use the properties of logarithms to find the exact value of the expression. Do not use a calculator. log 424 − log 46 A. 4 B. 24 C. 1 D. 6 Answer: Step 1 Given, log 4 24 − log 4 6 Step 2 Using product rule of logarithms. log 𝑚 + log 𝑛 = log(𝑚𝑛) Step 3 Apply the above rule log 4 24 − log 4 6 = log 4 (4  6) − log 4 6 = log 4 4 + log 4 6 = log 4 4 =1 Question #2 Solve the exponential equation 𝑒 4𝑥−5 − 7 = 11243. Express the solution set in terms of natural logarithms or common logarithms. Then use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. Answer: Step 1 To determine Given: 𝑒 4𝑥−5 − 7 = 11243 To determine: The solution in terms of logarithms & in decimal form. Step 2 Calculation Since, 𝑒 4𝑥−5 − 7 = 11243 ⇒ 𝑒 4𝑥−5 − 7 = 11243 + 7 𝑒 4𝑥−5 − 7 = 11250 Now, taking natural log on both the sides of equation, we get, ln(𝑒 4𝑥−5 ) = ln(11250) 4𝑥 − 5 ln(𝑒) = ln 1 1250 4𝑥 − 5 = ln(11250) 𝑥= ln(11250)+5 4 Hence, 𝑥 = ln(11250)+5 is the required solution in logarithm form. 4 Now, ln(11250) = 9.3281 So, 𝑥 = 9.3281+5 4 = 3.58 Step 3 Conclusion Hence, 𝑥 = ln(11250)+5 4 = 3.58 is the required solution Question #3 Review 1) 5𝑥−7 = 32𝑥+1 2) 11𝑒 −4𝑥 = 21 Answer: Step 1 To solve the equations for the value of x Step 2 1) Solution, using logarithms 1) 5𝑥−7 = 32𝑥+1 Take log an both sides: (𝑥 − 7) log 5 = (2𝑥 + 1) log 3 𝑥(log 5 − 2 log 3) = log 3 + 7 log 5 𝑥= log 3+7 log 5 log 5−2 log 3 = −21.036 Step 3 2) Solution, applying logarithms 2) 11𝑒 −4𝑥 = 21 ⇒ 𝑒 4𝑥 = 11 21 11 ⇒ 4𝑥 = ln ( ) 21 11 so, 𝑥 = 0.25 × ln ( ) = −0.1617 21 Question #4 Write the expression as a logarithm: 2 log 2 (𝑥 − 3) + log 2 (2𝑥 + 3) Answer: Step 1 Simplify the logarithmic expression as follows. 2 log 2 (𝑥 − 3) + log 2 (2𝑥 + 3) = log 2 ((𝑥 − 3)2 ) + log 2 (2𝑥 + 3) (∵ 𝑎𝑐 (𝑏) = log 𝑐 (𝑏 𝑎 )) Step 2 Use the product rule for logarithms. When adding the logs with same base, keep the log and base and multiply the elements as follows. 2 log 2 (𝑥 − 3) + log 2 (2𝑥 + 3) = log 2 ((𝑥 − 3)2 (2𝑥 + 3)) (∵ log 𝑐 (𝑎) + log 𝑐 (𝑏) = 𝑙𝑜𝑔𝑐 (𝑎𝑏)) Step 3 Answer: The expression can be written as log 2 ((𝑥 − 3)2 (2𝑥 + 3)) Question #5 DISCUSS: Comparing Logarithms Which is larger, log 4 17 or log 5 24 ? Explain your reasoning Answer: Step 1 Consider the logarithms, log 4 17 or log 5 24 Recall the same base rule of logarithm which states that, log 𝑏 = 1 and power rule of logarithm log 𝑏 𝑚𝑛 = 𝑛 𝑙𝑜𝑔𝑏 𝑚. Step 2 Here observe that log 4 17 > log 4 16 ⇒ log 4 17 > log 4 42  log 4 17  2 log 4 4 ⇒ log 4 17 > 2(1) ⇒ log 4 17 > 2 (same base rule ) Step 3 also the same way, one can see that log 5 2 4 < log 5 2  log 5 24  log 5 25  log 5 24  log 5 52  log 5 24  2 log 5 5  log 5 24  2(1) Step 4 (same base rule)  log 5 24  2 So, one can easily conclude that, log 4 17 > log 5 24 That is, log 4 17 is larger than log 5 24 Question #6 Write the expression as the sum or difference of two logarithmic functions containing no exponents. log ( 𝑥 𝑥+5 ) Answer: Step 1 To find an equlivalent expression in the required form Step 2 General fact about logarithms 𝑎 = 10𝑚 , 𝑏 = 10𝑛 , then (by defination). log 𝑎 = 𝑚, log 𝑏 = 𝑛 𝑎 Note that log ( ) = log(10𝑚−𝑛 ) 𝑏 = 𝑚 − 𝑛 = log 𝑎 − log 𝑏 𝑎 Thus, log ( ) = log 𝑎 − log 𝑏 𝑏 Step 3 Thus, the given expression has been expressed as the difference of two logarithms 𝑎 log ( ) = log 𝑎 − log 𝑏 𝑏 Thus, log ( 𝑥 𝑥+5 ) = log 𝑥 − log(𝑥 + 5) Question #7 1. Simplify the following expressions: a) 2 ln √𝑒 2 2 b) ln(𝑒 −𝑥 −𝑦 ) c) ln(ln 𝑒 𝑥 ) Answer: Step 1 First go Through the basic identity used in logarithms. On the right most side of solutions provided then go through the solution step by step you will definitely got the solutions. Step 2 1 a) 2 ln √𝑒 = 2 ln(𝑒)2 1 = 2 × ln(𝑒) 2 We know that ln 𝑎𝑏 = 𝑏 ln 𝑎 ln 𝑒 = 1 = 1 × 1 = 1 Answer 2 2 b) ln(𝑒 −𝑥 −𝑦 ) ⇒ (−𝑥 2 − 𝑦 2 ) ln 𝑒 ⇒ (−𝑥 2 − 𝑦 2 ) × 1 = −(𝑥 2 + 𝑦 2 ) Answer c) ln(ln 𝑒 𝑥 ) = ln[𝑥 ln 𝑒] = ln[𝑥 × 1] = ln 𝑥 Answer Question #8 Solve the logarithmic equation. Express irrational solutions in exact form and as a decimal rounded to three decimal places. 1 log(𝑥 − 1) = ( ) log 2 3 Answer: Step 1 The equation is a statement that consists of equal symbol between two algebraic expressions. The solution for the variable of the equation must satisfy the equation when we resubstitute the solution in the equation. The given equation involves logarithm terms. A logarithm is the power to which a number must be raised in order to get some other number. Example is if log 𝑎 (𝑥) = 𝑏 then 𝑥 = 𝑎𝑏 . Here a is the base of the logarithm. The common logarithms are logarithms with base 10 and these are represented with log(𝑥). Step 2 The given equation contains common logarithm on both sides. To solve the given equation solve using logarithm properties as follows; 1 log(𝑥 − 1) = ( ) log 2 ………Given equation. 3 1 log(𝑥 − 1) = log 23 ………..Use: If 𝑎 log(𝑥) = log(𝑥 𝑎 ) 1 𝑥 − 1 = 23 ……….. As base of both logarithms are same, equate the terms. 3 𝑥 − 1 = √2 3 𝑥 = 1 + √2 ………..Add both sides with 1. 3 𝑥 = 1 + √2 …………Solution in exact form. 3 Hence, the solution of the given equation asked in the question in exact form is 𝑥 = 1 + √2 Question #9 Use the properties of logarithms to expand the quantity. ln 𝑥2 (𝑦 3 𝑧 4 ) Answer: Step 1 It is given that, ln ( 𝑥2 𝑦3𝑧 4 ) Use the properties of logarithms, we have to expand the quantity. Step 2 We have, ln ( Now, ln ( ⇒ ln ( ⇒ ln ( 𝑥2 𝑦3 𝑧 4 𝑥2 𝑦3 𝑧 4 𝑥2 𝑦3𝑧 4 𝑥2 𝑦3 𝑧 4 ) 𝑚 ) = ln(𝑥 2 ) − ln(𝑦 3 𝑧 4 ) , {since ln ( ) = ln(𝑚) − ln(𝑛)} 𝑛 ) = ln(𝑥 2) − [ln(𝑦 3 ) + ln(𝑧 4 )], {since, ln(𝑚 × 𝑛) = ln(𝑚) + ln(𝑛)} ) = 2 ln(𝑥) − 3 ln(𝑦) − 4 ln(𝑧) , {since, ln(𝑚𝑎 ) = 𝑎 ln(𝑚)} Hence, ln ( 𝑥2 𝑦3𝑧 4 ) = 2 ln(𝑥) − 3 ln(𝑦) − 4 ln(𝑧) Question #10 Write the expression as a sum, difference, or product of logarithms. Assume that all variables represent positive real numbers: log 𝑏 ( 𝑚5 𝑝 9 𝑛3 𝑏 7 ) Answer: Step 1 Consider the provided logarithm expression, 𝑚5 𝑝9 log 𝑏 ( 3 7 ) 𝑛 𝑏 Step 2 The formula for logarithm is shown below, log _𝑏(𝑚𝑛) = log _𝑏𝑚 + log _𝑏𝑛 𝑚 log 𝑏 ( ) = log 𝑏 𝑚 − log 𝑏 𝑛 𝑛 logb (m)n = logb m log b b = 1 Step 3 Now, using the above formula to rewrite the above logarithm expression as, 𝑚5 𝑝 9 ) = log 𝑏 (𝑚5 𝑝9 ) − log(𝑛3 𝑏 7 ) [Difference Rule] 𝑛3 𝑏 7 log 𝑏 (𝑚5 ⋅ 𝑝9 ) − log 𝑏 (𝑛3 ⋅ 𝑏 7 ) [log 𝑏 (𝑚5 ) + log 𝑏 (𝑝9 )] − [log 𝑏 (𝑛3 ) + log 𝑏 (𝑏 7 )] [Product log 𝑏 (𝑚5 ) + log 𝑏 (𝑝9 ) − log(𝑛3 ) − log 𝑏 (𝑏 7 ) 5 log 𝑏 𝑚 + 9 log 𝑏 𝑝 − 3 log 𝑏 𝑛 − 7 log 𝑏 𝑏 [∵ log 𝑏 (𝑚)𝑛 log 𝑏 ( = = = = = 5 log 𝑏 𝑚 + 9 log 𝑏 𝑝 − 3 log 𝑏 𝑛 − 7 Rule] = 𝑛 log 𝑏 𝑚] Step 4 Answer: The provided logarithm expression can be written as sum, difference, or product of logarithms is, 𝑚5 𝑝9 log 𝑏 ( 3 7 ) = 5 log 𝑏 𝑚 + 9 log 𝑏 𝑝 − 3 log 𝑏 𝑛 − 7 𝑛 𝑏 Question #11 Compound interest Solve the compound interest formula 𝑟 𝑛𝑡 𝐴 = 𝑃 (1 + ) 𝑛 for t by using natural logarithms Answer: Step 1 Given that the compound interest formula, 𝑟 𝑛𝑡 𝐴 = 𝑃 (1 + ) … … … . (1) 𝑛 Step 2 Take the natural logarithms both sides, 𝑟 𝑛𝑡 ln(𝐴) = ln [(1 + ) ] 𝑛 𝑟 𝑛𝑡 ln(𝐴) = ln(𝑃) + ln (1 + ) 𝑛 𝑟 ln(𝐴) − ln(𝑃) = 𝑛𝑡 ⋅ ln (1 + ) 𝑛 𝑛𝑡 = 𝑡= ln(𝐴)−ln(𝑃) 𝑟 𝑛 ln(1+ ) 1 ln(𝐴) − ln(𝑃) ⋅ 𝑛 ln (1 + 𝑟 ) 𝑛 Question #12 Use the properties of logarithms to expand the following expression as much as possible. Simplify any numerical expressions that can be evaluated without a calculator. log 4 (48𝑥 + 16𝑦) Answer: Step 1 Given The following expression: log 4 (48𝑥 + 16𝑦) Step 2 To determine To expand the following expression using the properties of logarithms. Let, log 4 (48𝑥 + 16𝑦) Expanding the above equation, we get log 4 (48𝑥 + 16𝑦) = = = = log(48𝑥+16𝑦) log(4) − − − −{∵ by Change of base formula} log(16)(3𝑥+𝑦) log(4) log(16)+log(3𝑥+𝑦) − − − −{∵ log 𝑏 (𝑚𝑛) = log 𝑏 (𝑚) + log 𝑏 (𝑛)} log(4) log(16) log(3𝑥+𝑦) log(4) + log(4) = log 4 (42 ) + log 4 (3𝑥 + 𝑦) − − − −{∵ log 𝑏 (𝑥) = log𝑑 (𝑥) } log𝑑 (𝑏) = 2 log 4 (4) + log 4 (3𝑥 + 𝑦) = 2 + log 4 (3𝑥 + 𝑦) − − − −{∵ log 𝑏 (𝑏) = 1} Step 3 Therefore, log 4 (48𝑥 + 16𝑦) = 2 + log 4 (3𝑥 + 𝑦) Question #13 Write the expression as a single logarithm 6 log 5 (𝑥 − 3) − 4 log 5 (𝑥 + 4) Answer: Step 1 To express the given quantity as the logarithm of a single expression Step 2 Recall the basic facts regarding logarithms. All logs are to the same base (say 5, as in this problem) 𝑚 log 𝑥 = log 𝑥 𝑚 𝑝 log = log 𝑝 − log 𝑞 𝑞 Step 3 𝑚 log 𝑥 = log 𝑥 𝑚 So, 6 log(𝑥 − 3) = log(𝑥 − 3)6 4 log(𝑥 + 4) = log(𝑥 + 4)4 Step 4 ANSWER: GIven expression has been expressed as the log (to base 5) of a single term 𝑝 log = log 𝑝 − log 𝑞 𝑞 So, log(𝑥 − 3)6 − log(𝑥 + 4)4 = log (𝑥−3)6 (𝑥+4)4 Question #14 The product property of logarithms states that log 𝑏 (𝑥𝑦) = for positive real numbers b,x and where 𝑏 ≠ 1 Answer: Step 1 The product property of logarithms states that log 𝑏 (𝑥𝑦) = log 𝑏 𝑥 + log 𝑏 𝑦 for the positive real number b , x and y where Step 2 Solution: The product property of logarithms states that log b ( xy ) = log b x + log b y for the positive real number b, x and y where 𝑏 ≠ 1 Question #15 Ise properties of logarithms to expand the logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator log 5 ( 625 𝑦 ) Answer: Step 1 According to the properties of logarithms, we have the following: 𝑏 log 𝑎 ( ) = log 𝑎 (𝑏) − log 𝑎 (𝑐) and log 𝑎 (𝑎𝑛 ) = 𝑛 𝑐 Step 2 The given logarithmic expression can be simplified as shown below: log 5 ( 625 𝑦 ) = log(625) − log 5 (𝑦) = log 5 (54 ) − log 5 (𝑦) = 4 − log 5 (𝑦) Question #16 Solve, 3 log 8 0 − ln 5 log 5 Answer: Step 1 Consider the provided expression. 3 log 8 0 − ln 5 log 5 Now, simplifying the above expression using the property of logarithms. 3 log10 80 − (ln 5) log10 5 = log10 (80)3 − log10 (5)ln 5 (80)3 = log10 ((5)ln 5) Step 2 further simplify, 4096 3 log10 80 − (ln 5) log10 5 = log10 ((5)ln 5) = log10 (4096 ⋅ 5− ln(5)+3 ) ≈ 5.48 Question #17 {𝐶𝑜𝑛𝑑𝑒𝑛𝑠𝑒 𝑡ℎ𝑒 𝑓𝑜𝑙𝑙𝑜𝑤𝑖𝑛𝑔 𝑒𝑥𝑝𝑟𝑒𝑠𝑠𝑖𝑜𝑛 𝑖𝑛𝑡𝑜 𝑎 𝑠𝑖𝑛𝑔𝑙𝑒 𝑙𝑜𝑔𝑎𝑟𝑖𝑡ℎ𝑚 𝑢𝑠𝑖𝑛𝑔 𝑡ℎ𝑒 𝑝𝑟𝑜𝑝𝑒𝑟𝑡𝑖𝑒𝑠 𝑜𝑓 𝑙𝑜𝑔𝑎𝑟𝑖𝑡ℎ𝑚𝑠. 𝐴𝑠𝑠𝑢𝑚𝑒 𝑥 > 0 𝑎𝑛𝑑 𝑥 ≠ 6 log 5 (𝑥 2 − 8𝑥 + 12) − 2 log 5 (𝑥 − 6) Answer: Step 1 According to the definition of logarithm, if 𝑦 = 𝑎 𝑥 , then log 𝑎 (𝑦) = 𝑥 , where a is called the base of the logarithm. According to the base changing rule, logarithm of any number to the base a can be converted into another, that is log 𝑎 (𝑦) = log𝑏 (𝑦) log𝑏 (𝑎) If the base of the logarithm is 10, then it is called the common logarithm, example log10 (𝑦) , log10 (102) etc. Similarly if the base of the logarithm is e, then it is called natural logarithm, examples log 𝑒 (𝑦) , log 𝑒 (102.25) 𝑒𝑡𝑐. Step 2 To express log 5 (𝑥 2 − 8𝑥 + 12) − 2 log 5 (𝑥 − 6) as a single logarithm consider the following steps. ⋆ According to the power property of logarithm, the logarithm of the exponent of a number (𝑝𝑞 ) is equal to the exponent times the logarithm of the number (p), that is, 𝑞 log 𝑎 (𝑝) = log 𝑎 (𝑝𝑞 ). Use power property to express 2 log 5 (𝑥 − 6) as log 5 (𝑥 − 6)2 ⋆ According to the quotient property of logarithm, the logarithm of the quotient of two numbers is equal to 𝑚 the difference of their individual logarithms, that is, log 𝑎 (𝑚) − log 𝑎 (𝑛) = log 𝑎 ( ). In this problem a=5, 𝑛 𝑚 = 𝑥 2 − 8𝑥 + 12 and 𝑛 = (𝑥 − 6)2 ⋆ The expression 𝑥 2 − 8𝑥 + 12 can be written as (x-6)(x-2). To simplify the quantity inside the parenthesis of log 5 [ log 5 (𝑥 2 (𝑥 2 −8𝑥+12) (𝑥−6)2 ], cancel out the common x-6 factor. − 8𝑥 + 12) − 2 log 5 (𝑥 − 6) = log 5 (𝑥 2 − 8𝑥 + 12) − log 5 (𝑥 − 6)2 (𝑥 2 −8𝑥+12) = log 5 [ ] = log 5 [ ] (𝑥−6)2 (𝑥−6)(𝑥−2) (𝑥−6)2 (𝑥−2) = log 5 [(𝑥−6)] Question #18 Solve the logarithmic equation ln(𝑥 − 4) + ln(𝑥 + 1) = ln(𝑥 − 8). Be sure to reject any value of x that is not in the domain of the original logarithmic expressions. Give the exact answer. Then, where necessary, use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. Answer: Step 1 Given ln(𝑥 − 4) + ln(𝑥 + 1) = ln(𝑥 − 8) Step 2 It is given that, ln(𝑥 − 4) + ln(𝑥 + 1) = ln(𝑥 − 8) On rewriting the natural logarithm using the base e. log 𝑒 (𝑥 − 4) + log 𝑒 (𝑥 + 1) = log 𝑒 (𝑥 − 8) On expressing the given equation in the form log 𝑏 𝑀 = 𝑐 Applying the product rule log 𝑏 (𝑀𝑁) = log 𝑏 𝑀 + log 𝑏 𝑁on the left side. log 𝑒 [(𝑥 − 4)(𝑥 + 1)] = log 𝑒 (𝑥 − 8) Step 3 It is known that if log 𝑏 𝑀 = log 𝑏 𝑁 then M=N On using the property to rewriting the equation without logarithms. (𝑥 − 4)(𝑥 + 1) = (𝑥 − 8) Simplifying the left side, 𝑥 2 − 3𝑥 − 4 = 𝑥 − 8 On rewriting the equation in the standard form 𝑥 2 − 3𝑥 − 4 − 𝑥 + 8 = 𝑥 − 8 − 𝑥 + 8 𝑥 2 − 4𝑥 + 4 = 0 Step 4 On factorizing 𝑥 2 − 4𝑥 + 4 (x-2)(x-2)=0 On applying the zero-product principle x-2=0 x=2 Step 5 On checking the proposed solutions in the original equation, ln(𝑥 − 4) + ln(𝑥 + 1) = ln(𝑥 − 8) ln(2 − 4) + ln(2 + 1) = ln(2 − 8) ln(−2) + ln 3 = ln(−6) The given statement is false Since the negative numbers do not have logarithms the solution 2 does not check and the given equation has no solution therefore the solution set is the empty set ϕ. Question #19 Use the Properties of Logarithms to condense the logarithm. Simplify if possible. (Assume the variable is positive.) log 3 (𝑥 2 − 1) − 2 log 3 (𝑥 − 1) Answer: Step 1 To simplify: log 3 (𝑥 2 − 1) − 2 log 3 (𝑥 − 1) by the use of Properties of Logarithms. Formula used: Logarithmic properties. 1. 𝑎 log 𝑐 (𝑏) = log 𝑐 (𝑏 𝑎 ) 𝑎 2. log 𝑐 (𝑎) − log 𝑐 (𝑏) = log 𝑐 ( ) 𝑏 Step 2 Calculation: Apply log rule 𝑎 log 𝑐 (𝑏) = log 𝑐 (𝑏 𝑎 ) log 3 (𝑥 2 − 1) − 2 log 3 (𝑥 − 1) = log 3 (𝑥 2 − 1) − log 3 ((𝑥 − 1)2 ) 𝑎 Now apply the log rule log 𝑐 (𝑎) − log 𝑐 (𝑏) = log 𝑐 ( ) 𝑏 𝑥 2 −1 log 3 (𝑥 2 − 1) − log 3 ((𝑥 − 1)2 ) = log 3 ((𝑥−1)2 ) 𝑥 2 −1 log 3 (𝑥 2 − 1) − log 3 ((𝑥 − 1)2 ) = log 3 ((𝑥−1)2 ) = log 3 ( = log 3 ( (𝑥+1)(𝑥−1) (𝑥−1)2 𝑥+1 𝑥−1 ) ) Step 3 Answer: log 3 (𝑥 2 − 1) − log 3 ((𝑥 − 1)2 ) = log 3 ( Question #20 Solve log 4 𝑥 = Answer: 5 2 𝑥+1 𝑥−1 ) Step 1 We have to solve the equation: 5 log 4 𝑥 = 2 We know the properties of logarithms, log 𝑎 (𝑏) = 𝑐 ⇒ 𝑏 = 𝑎𝑐 Step 2 Applying above property for the given equation logarithm, 5 log 4 𝑥 = 2 5 𝑥 = 42 5 = 22×2 = 25 = 2×2×2×2×2 =32 Hence, solution of the equation is x=32.