Title: M4.2 Exponential and Logrithmic Quiz
Course: Technical Math for Industry > Section: Module 4.2 Exponential and Logarithmic Functions

Q: Which of the following could be the graph of ?
A: (a choice with no text)
Feedback: Incorrect. This is the graph of a function in the form , where . In the given function, .
Feedback: Incorrect. The function is in the form , where . You may have thought that because b is between 0 and 1, the graph would be inverted.
Feedback: Incorrect. The function is in the form , where . You may have thought that because b is between 0 and 1, the graph would be inverted.
Feedback: Correct. Since the function is in the form , where , its graph will decrease from left to right.

Q: Investing $12,000 in a savings account at 2% annual interest compounded semiannually will result in approximately how much money after 5 years?

Use the formula:
- $12,865.62 (Incorrect. You may have added the 2 and 5 to obtain 7 as the exponent. Be sure to multiply those two numbers and follow the correct order of operations.)
- $13,200.00 (Incorrect. It appears that you found the simple interest instead of using the given formula. You are given , , , and . Substitute these values into the formula and simplify.)
- $13,255.47 (Correct. You are given , , , and . Substituting into the formula gives you: So .)
- $13,248.97 (Incorrect. You may have missed the ''semi'' part of ''semiannually,'' and used . You should be using , along with , , and . Substitute into the formula and simplify.)
A: $13,255.47

Q: Convert into an exponential equation.
A: (a choice with no text)
Feedback: Incorrect. You have confused the values of the base b and the exponent y. Remember that the base b is the number written below the line in the logarithmic equation. In this case, .
Feedback: Incorrect. Although this is a true equation, it is not the exponential form of the given equation, in which the base , the exponent , and .
Feedback: Incorrect. This is not a true equation. Remember that the base b is the number written below the line in the logarithmic equation. In the given equation, .
Feedback: Correct. The base , the exponent , and , so .

Q: Rewrite as a logarithmic equation.
A: (a choice with no text)
Feedback: Incorrect. You have mixed up the b value and the y value. Remember that is equivalent to . Identify the values of the variables in the exponential equation and substitute them into the logarithmic equation.
Feedback: Correct. The equation is equivalent to , so , , and . Substitute these values to obtain:
Feedback: Incorrect. Remember that is equivalent to . Identify the values of the three variables in the exponential equation and substitute them into the logarithmic equation.
Feedback: Incorrect. Remember that is equivalent to . Identify the values of the three variables in the exponential equation and substitute them into the logarithmic equation.

Q: Which graph could be the graph of ?
A: (a choice with no text)
Feedback: Incorrect. This is the right general shape, but it is the graph of .
Feedback: Correct. The graph passes through , , and , for example. These numbers make the equation true. In general, when , the graph of is increasing and has this shape.
Feedback: Incorrect. You may have reversed the x and y values in your table of values. The equation is represented by this graph.
Feedback: Incorrect. This is the graph of . Be sure to note that, for the function in the question, the base , so the graph should be increasing.

Q: Solve:
A: (a choice with no text)
Feedback: Correct. is equivalent to . Try different bases: , , So .
Feedback: Incorrect. It appears that you mixed up the b value and the x value. Remember that is equivalent to . Try different bases until you find a number to the 4th power equal to 10,000.
Feedback: Incorrect. It appears that you divided 10,000 by 4. Remember that is equivalent to . Try different bases until you find a number to the 4th power equal to 10,000.
Feedback: Incorrect. You found the solution to . Remember that is equivalent to . Try different bases until you find a number to the 4th power equal to 10,000.

Q: Simplify
A: (a choice with no text)
Feedback: Incorrect. You may have separated the logarithm into two logarithms, but then multiplied instead of adding them. Use the product property, . Then simplify.
Feedback: Correct. Use the product property:
Feedback: Incorrect. You found , which is correct, but you did not use the product property, . Use the product property and then simplify.
Feedback: Incorrect. You may have found the value of to be 3, which is correct, but you did not use the product property, . Use the product property and then simplify.

Q: Use the quotient property to rewrite
A: (a choice with no text)
Feedback: Incorrect. This is not equivalent to the original expression. The quotient property says: Use this on the given expression and then simplify.
Feedback: Incorrect. It appears that you reversed terms when using the quotient property. Start with and then simplify.
Feedback: Correct. Use the quotient property, , and then simplify, using the fact that .
Feedback: Incorrect. You have found an equivalent expression, but can still be simplified.

Q: Simplify:
A: (a choice with no text)
Feedback: Incorrect. You have found an equivalent expression, but can be simplified further.
Feedback: Incorrect. You may have remembered that when you use the power property, the exponent becomes a factor. However, it should multiply the logarithm. The expression simplifies to . You will then need to simplify .
Feedback: Incorrect. You may have correctly simplified to , but then incorrectly simplified . Use the definition of logarithm.
Feedback: Correct. Use the power property: This simplifies to . Because , simplifies to 3. So .

Q: Rewrite using the properties of logarithms.
A: (a choice with no text)
Feedback: Correct. First use the power property, then use the quotient property:
Feedback: Incorrect. You probably used the power property correctly, but then misused the quotient property. Remember that .
Feedback: Incorrect. You may have simplified correctly for the first two steps, but then made a mistake with the factor 3. Start with the power property, then the quotient property: If you have a term such as , you may not rewrite it with the factor 3 inside the logarithm.
Feedback: Incorrect. You may have incorrectly rewritten the logarithm as . You must first use the power property, and then use the quotient property.

Q: If you enter 5 into a scientific calculator and press the ln key, you would be finding the value of which of the following expressions?

(On a graphing calculator, to do this same calculation, you press the ln key, then 5, then Enter.)
A: (a choice with no text)
Feedback: Correct. The ln key is used to find the natural logarithm. Because the natural logarithm of a number uses base e, the expression you are finding can be written as .
Feedback: Incorrect. You mixed up the placement of the values 5 and e in the expression. The ln key is used to find the natural logarithm. This is a logarithm with a base of e.
Feedback: Incorrect. You may have confused natural and common logarithms, and mixed up the placement of the values in the expression. The ln key is used to find the natural logarithm. This is a logarithm with a base of e.
Feedback: Incorrect. You have confused natural and common logarithms. The ln key is used to find the natural logarithm, so you are finding the natural logarithm of 5, not the common logarithm.

Q: Which function is represented by the graph below?
A: (a choice with no text)
Feedback: Incorrect. You may have confused the shape of exponential and logarithmic graphs. The general shape of a logarithmic graph is this: it gets close to the y-axis for x near 0, increases as x increases, and is curved downward.The given graph does not have this shape.
Feedback: Incorrect. You may have chosen this answer because you are not familiar with the graphs of exponential functions, or you misread the answer. The graph of would be linear, because the variable x is being multiplied by two constants, e and 2. The given graph is not a straight line.
Feedback: Correct. The graph gets close to the x-axis on the left, and gets steeper and steeper on the right, which means that it has the shape of an exponential graph. Because the graph passes through points (0, 1) and (0.5, 2.7. . .), it matches the graph of .
Feedback: Incorrect. You may have confused the shape of the graphs of and . The general shape of a logarithmic graph is this: it gets close to the y-axis for x near 0, increases as x increases, and is curved downward. The given graph does not have this shape.

Q: Use the values and to find the approximate value of .
A: (a choice with no text)
Feedback: Incorrect. You added the approximate values of the natural logarithms. Instead, you should have used the Change of Base formula to rewrite the logarithm as
Feedback: Correct. Use the Change of Base formula to rewrite using natural logarithms: Use the given values to approximate the quotient:
Feedback: Incorrect. You multiplied the approximate values of the natural logarithms. Instead, you should have used the Change of Base formula to rewrite the logarithm as
Feedback: Incorrect. You probably mistakenly found , instead of

Q: Solve:
A: (a choice with no text)
Feedback: Incorrect. You may have correctly set the exponents equal to each other and found that , but then you evaluated the expressions on each side of the equation for and found . This is not the value of x.
Feedback: Incorrect. You probably rewrote the equation as , which is true, but then incorrectly set the exponents equal to each other and solved for x. Those exponents are not equal because the bases are not equal. Instead, rewrite the equation as , and then set the exponents equal.
Feedback: Correct. Evaluate : Because both exponential expressions have the same base, their exponents must be equal. Write a new equation that sets the exponents equal to each other: Solve the linear equation as you normally would:
Feedback: Incorrect. You may have thought that both exponential expressions having a base of 9 would make the value of x. However, because the bases are the same, you need to set the exponents equal to each other and solve for x.

Q: Solve:
A: (a choice with no text)
Feedback: Incorrect. It appears that you used the product property to get . Instead, you should use the quotient property to simplify .
Feedback: Incorrect. You added to both sides of the equation, and then simplified to , instead of .
Feedback: Incorrect. You may have correctly rewritten the equation as . But it appears that you tried to simplify by dividing both sides by . However, is not the same as . The expression tells you to multiply x by 8 and then compute a logarithm. So is not a factor of .
Feedback: Correct. Use the quotient property on the left side: The equation becomes , so Divide both sides by 4:

Q: An earthquake near New Zealand measured 4.2 on the Richter scale. Use the formula to determine approximately how many times stronger the wave amplitude of the earthquake was than .
A: (a choice with no text)
Feedback: Correct. Substitute 4.2 for R in the formula: Rewrite as an exponential equation: Use a calculator and multiply both sides by :
Feedback: Incorrect. You found . However, 4.2 is the value of R. Substitute correctly into the formula and then rewrite it as an exponential equation.
Feedback: Incorrect. You may have correctly substituted 4.2 for R in the formula, but then treated the equation as if it had a natural logarithm, instead of a common logarithm.
Feedback: Incorrect. You may have found the correct factor, but then multiplied this amount by the original Richter scale measurement of 4.2 and gave this amount as the answer.

Q: The population of a town can be modeled using the formula , where t is the number of years after 2012 and P is the town's population. Which of the following equations can be used to find the number of years after 2012 that the population will double to 40,000?
A: (a choice with no text)
Feedback: Incorrect. You may have used a common logarithm on one side of the equation and a natural logarithm on the other. In general, you must perform the same operation on both sides of an equation. In particular, you must use logarithms with the same base on both sides of the equation when solving for a variable.
Feedback: Incorrect. You may have correctly started with , and then subtracted 20,000 instead of dividing both sides by 20,000.
Feedback: Incorrect. You correctly substituted the value 40,000 into the equation and divided both sides by 20,000 to get . However, you incorrectly divided both sides by while trying to isolate the variable. You should instead take logarithms of both sides.
Feedback: Correct. Substitute the value 40,000 in the formula: Divide both sides by 20,000: Take logarithms of both sides of the equation: Because , we have , and therefore Divide both sides by 0.02 to find t: