Title: Unit 18 Quiz
Course: Technical Math for Industry

Q: Which function could be represented by this graph?
A: (a choice with no text)
If right: Correct. The points , , and are solutions to the equation and are located on the given graph.
If wrong: Incorrect. This is another exponential growth function and has the same shape as the given graph, but it is not an exact match. For example, this function would pass through the points and , and these points are not on the given graph.
If wrong: Incorrect. The graph for this function would show exponential decay. That is, it would descend from left to right.
If wrong: Incorrect. You may have chosen this equation because its graph passes through . However, the graph of is a straight line.

Q: Investing $5,000 in a savings account at 1% annual interest compounded quarterly will result in approximately how much money after 2 years?

Use the formula:
- $5,100.50 (Incorrect. You may have missed the word quarterly and used . You should be using , along with , , and . Substitute into the formula and simplify.)
- $5,100.00 (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.)
- $5,075.47 (Incorrect. You may have added the 4 and 2 to obtain 6 as the exponent. Be sure to multiply those two numbers and follow the correct order of operations.)
- $5,100.88 (Correct. You are given , , , and . Substituting into the formula gives you: So .)
A: $5,100.88

Q: Rewrite as an exponential equation.
A: (a choice with no text)
If wrong: Incorrect. Although this is a true equation, it is not the exponential form of the given equation. Remember that is equivalent to .
If wrong: Incorrect. Although this is a true equation, it is not an exponential equation. The base , the exponent , and .
If right: Correct. The base , the exponent , and , so .
If wrong: 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, .

Q: Convert into a logarithmic equation.
A: (a choice with no text)
If wrong: Incorrect. You have confused the base and the exponent. Remember that is equivalent to . Identify the values of the variables in the exponential equation and substitute them into the logarithmic equation.
If wrong: Incorrect. You have confused the x and y values. Remember that is equivalent to . Identify the values of the variables in the exponential equation and substitute them into the logarithmic equation.
If wrong: Incorrect. Although this is a true equation, it is not the logarithmic form of the given equation. Remember that is equivalent to . Identify the values of the variables in the exponential equation and substitute them into the logarithmic equation.
If right: Correct. The equation is equivalent to , so: , , and Substitute these values into the logarithmic equation and you have:

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

Q: Solve:
A: (a choice with no text)
If wrong: Incorrect. It appears that you divided 400 by 2. Remember that is equivalent to . Try different bases until you find a number to the 2nd power equal to 400.
If wrong: Incorrect. It appears that you multiplied 2 by 400. Remember that is equivalent to . Try different bases until you find a number to the 2nd power equal to 400.
If wrong: 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 2nd power equal to 400.
If right: Correct. is equivalent to . Try different bases: , , So .

Q: Use the product property to rewrite .
A: (a choice with no text)
If wrong: Incorrect. You may have separated the logarithm into two logarithms, but then multiplied instead of adding them. Use the product property, . Then simplify.
If wrong: Incorrect. You may have found the value of to be 1, which is correct, but you did not use the product property, . Use the product property and then simplify.
If wrong: Incorrect. You may have found the value of to be 1, which is correct, but you did not use the product property, . Use the product property and then simplify.
If right: Correct. Use the product property:

Q: Simplify:
A: (a choice with no text)
If wrong: Incorrect. You may have realized that , but you need to use the quotient property first:
If wrong: Incorrect. You may have realized that , but you need to use the quotient property first:
If wrong: Incorrect. You have correctly used the quotient property, but this can be simplified further. Rewrite 216 as a power of 6 and use the definition of logarithm.
If right: Correct. Use the quotient property: Then use the fact that .

Q: Simplify:
A: (a choice with no text)
If wrong: Incorrect. You may have correctly simplified to , but then incorrectly simplified . Use the definition of logarithm.
If wrong: Incorrect. You have found an equivalent expression, but can be simplified further.
If wrong: Incorrect. You may have remembered that when you use the power property, the exponent becomes a factor. However, the exponent should be multiplied by the logarithm. The expression simplifies to . You will then need to simplify .
If right: Correct. Use the power property: This simplifies to . Because , simplifies to 2. So .

Q: Rewrite using the properties of logarithms.
A: (a choice with no text)
If wrong: Incorrect. You may have incorrectly rewritten the logarithm as . You must first use the power property, and then use the quotient property.
If wrong: Incorrect. You may have simplified correctly for the first two steps, but then made a mistake with the factor 2. Start with the power property, then the quotient property: If you have a term such as , you may not rewrite it with the factor 2 inside the logarithm.
If right: Correct. First use the power property, then use the quotient property:
If wrong: Incorrect. You probably used the power property correctly, but then misused the quotient property. Remember that .

Q: If you enter 7 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 7, then Enter.)
A: (a choice with no text)
If right: 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 .
If wrong: Incorrect. You have confused common and natural logarithms. The ln key is used to find the natural logarithm, so you are finding the natural logarithm of 7, not the common logarithm.
If wrong: Incorrect. You mixed up the placement of the values 7 and e in the expression. The ln key is used to find the natural logarithm. This is a logarithm with a base of e.
If wrong: 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.

Q: Which function is represented by the graph below?
A: (a choice with no text)
If right: Correct. The graph gets closer and closer to the y-axis, increases as the input increases, and curves downward. This is the general shape of the graph of any logarithm. The given graph passes through . You can check without using a calculator to see that this is a point on the graph of . Other points can be checked with a calculator.
If wrong: Incorrect. You may have chosen this answer because you are not familiar with the graphs of logarithmic functions. The graph of would be a straight line, because the variable x is being multiplied by two constants, e and 0.5. The given graph is not a straight line.
If wrong: Incorrect. You may have recognized that the general shape of the graph matches the graph of a logarithm. To see that it is not an exact match, you need to use a calculator. For example, . However, for the given graph, when , y is close to 1.6.
If wrong: Incorrect. You may have confused the general shapes of the graphs of logarithms and exponential functions. You can see, for example, that , so the point would be on the graph of . However, the point is not on the graph shown above.

Q: Use the values and to find the approximate value of .
A: (a choice with no text)
If right: Correct. Use the Change of Base formula to rewrite using natural logarithms: Use the given values to approximate the quotient:
If wrong: Incorrect. You probably mistakenly found , instead of
If wrong: 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
If wrong: 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

Q: Solve:
A: (a choice with no text)
If wrong: 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 resulting equation for and found and . This is not the value of x.
If wrong: 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.
If right: 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:
If wrong: Incorrect. You may have thought that both exponential expressions having a base of 32 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)
If wrong: Incorrect. It appears that you attempted to simplify the left side of the equation to . However, is not the same as . Instead, you should use the product property to simplify .
If wrong: Incorrect. You subtracted from both sides of the equation, and then simplified to , instead of .
If right: Correct. Use the product property on the left side: The equation becomes: , so Divide both sides by 12:
If wrong: 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 4 and then compute a logarithm. So is not a factor of .

Q: An earthquake in California measured 3.6 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)
If wrong: Incorrect. You may have found the correct factor, but then multiplied this amount by the original Richter scale measurement of 3.6 and gave this amount as the answer.
If right: Correct. Substitute 3.6 for R in the formula: Rewrite as an exponential equation: Use a calculator and multiply both sides by :
If wrong: Incorrect. You found . However, 3.6 is the value of R. Substitute correctly into the formula and then rewrite it as an exponential equation.
If wrong: Incorrect. You may have correctly substituted 3.6 for R in the formula, but then treated the equation as if it had a natural logarithm, instead of a common logarithm.

Q: The population of a city can be modeled using the formula , where t is the number of years after 2012 and P is the city's population. Which of the following equations can be used to find the number of years after 2012 that the population will triple to 300,000?
A: (a choice with no text)
If wrong: 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.
If wrong: Incorrect. You correctly substituted the value 300,000 into the equation and divided both sides by 100,000 to get . However, you incorrectly divided both sides by while trying to isolate the variable. You should instead take logarithms of both sides.
If wrong: Incorrect. You may have correctly started with , and then subtracted 100,000 instead of dividing both sides by 100,000.
If right: Correct. Substitute the value 300,000 in the formula: Divide both sides by 100,000: Take logarithms of both sides of the equation: Because , we have , and therefore Divide both sides by 0.05 to find t: