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Lesson 4: Logarithmic Functions TEST I: Evaluation DIRECTIONS: Evaluate the following equations. Show your complete solution. (3 points each) 1. log,> =y Step 1: Remove the fraction — [log] = y Step 2: Apply Identity Rule -l=y FINAL ANSWER: y= 71 2. logx + log(x — 1) = log(4x) Step 1: Same base, apply log rules (x)(x — 1) = 4x Step 2: Evaluate value of x (x) - 1) -— 4x =0 x —x-—4x=0 x — 5x =0 x(x - 5) =0 x=0 | x-5=0 x=0&x=5 Step 3: Checking log (0) = undefined FINAL ANSWER: x=2 3. Inx =— 3 Step 1: Definition of Natural Logarithm Any natural logarithm has base e In,(x) = -3 Step 2: Transform to Exponential Form -3 e =x 1 x=z e FINAL ANSWER: 1 x= e 4. 2logx = log2 + log(3x — 4) Step 1: Apply power property and multiplication property v= 2(3x — 4) Step 2: Solve for x x = 2(3x - 4) x = 6x -8 x —6x+8=0 (x — 4) — 2) =0 x-4=0 | x-2=0 x=4& x=2 Step 3: Checking Both are solutions to the given equation FINAL ANSWER: x=4andx=2 TEST II: Problem Solving DIRECTIONS: Solve the following equations. Show your complete solution. (5 points each) 6. Suppose that Juni invests $10000 in an investment earning 5% per year. He wants to know how long it would take his investment to accumulate to $12000, and how long it would take to accumulate to $15000. Step 1: We start by writing the exponential growth function that models the value of this investment as a function of the time since the $10000 is initially invested. y = 10000(1. 05)’ Step 2: We divide both sides by 10000 to isolate the exponential expression on one side. a t 10000 ~~ 1.05 Step 3: Next, we rewrite this in logarithmic form to express time as a function of the accumulated future value. We'll use function notation and call this function g(y). t= 9) = log, (saan) Step 4: Use the change of base formula to express t asa function of y using natural logarithm: _ In| “To000. t = gO)Tnt05 Step 5: We can now use this function to answer Juni’s questions. To find the number of years until the value of this investment is $12,000, we substitute y = $12, 000, into function g and evaluate t: - in(o000) __tn.2) t = g(12000). TaCL.05) = Tn(.05) = 3.74 years To find the number of years until the value of this investment is $15,000, we substitute y = $15,000 into function g and evaluate t: = Ison) Ins) t = g(15000): TatL.05) = Ino) = 8.31 years FINAL ANSWER: It will take 3.74 years until the value of Juni’s investment is $12,000. It will take 8.31 years until the value of Juni’s investment is $15,000. 7. One hot water pump has a noise rating of 50 decibels. One dishwasher, however, has a noise rating of 62 decibels. The dishwasher noise is how many times more intense than the hot water pump noise? Step 1: You can't easily compare the two noises using the formula, but you can compare them to P,,. Start by finding the intensity of noise for the hot water pump. Use h for the intensity of the hot water pump's noise. 50 = 10106(-¢} oO Step 2: Divide the equations by 10 to get the log by itself. = ah 5= too P, Step 3: Rewrite the equation as an exponential equation. 10° = Py Step 4: Multiply by P, to get h by itself. 5 h=10 Py Step 5: Repeat the same process to find the intensity of the noise for the dishwasher. 62 = 1010(-¢] 0 6.2 = tool 4] 62d 10" => 6.2 d=10 Py Step 6: To compare d to h, you can divide. (Think: if the dishwasher’s noise is twice as intense as the pump’s, then d should be 2h—that is, <4 should be 2.) Use the laws of exponents to simplify the quotient. 6.2 10 Py a h _~ 5. 10 Py 1.2 = 10 
