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Chapter 15: Problem 140

The base- 10 logarithm of 60 is a number: (a) Between \(-2\) and \(-1\) (b) Between \(-1\) and 0 (c) Between 0 and 1 (d) Between 1 and 2 (e) Between 2 and 3

Short Answer

Expert verified
The base-10 logarithm of 60 is a number between 1 and 2 (option (d)).

Step by step solution

01

Analyze the given ranges of the values

We will test each of the given ranges in the problem to find the appropriate range where the base-10 logarithm of 60 might fall. To do this, we will raise 10 to the power of the lower and upper limits of each range and compare it to 60.
02

Test Range (a): Between -2 and -1

In this range, we test the values -2 and -1 as the exponents: 10^{-2} = 0.01 \\ 10^{-1} = 0.1 Since neither of these equal 60, or even close, we move on to the next range.
03

Test Range (b): Between -1 and 0

Here, we test the values -1 and 0 as the exponents: 10^{-1} = 0.1 \\ 10^{0} = 1 Since 60 is not between 0.1 and 1, we can exclude this range.
04

Test Range (c): Between 0 and 1

In this range, we test the values 0 and 1: 10^{0} = 1 \\ 10^{1} = 10 As 60 is between 1 and 10, this range is a candidate. However, we will also check the other two ranges just to confirm.
05

Test Range (d): Between 1 and 2

In this range, we test the values 1 and 2: 10^{1} = 10 \\ 10^{2} = 100 Since 60 is between 10 and 100, this range is also a candidate.
06

Test Range (e): Between 2 and 3

Now, we test the last range with values 2 and 3: 10^{2} = 100 \\ 10^{3} = 1000 As 60 is not between these numbers, we can eliminate this range.
07

Conclusion

We found that the base-10 logarithm of 60 falls in the following two ranges: - Between 0 and 1 (10^0 to 10^1, or 1 to 10) - Between 1 and 2 (10^1 to 10^2, or 10 to 100) In this problem, we are looking for the exponent to which we raise 10 to get 60. Since 60 is between 10 and 100, but closer to 100, its base-10 logarithm must be: (d) Between 1 and 2.

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Most popular questions from this chapter

True or false? In an aqueous solution at \(25^{\circ} \mathrm{C}\), you will always get the same number when you multiply the equilibrium \(\mathrm{H}_{3} \mathrm{O}^{+}\) concentration by the equilibrium OH \(^{-}\) concentration. Explain your answer.

Ethanol, \(\mathrm{C}_{2} \mathrm{H}_{5} \mathrm{OH}\), and hydride ion, \(\mathrm{H}^{-}\), react to produce \(\mathrm{H}_{2}\) gas and the \(\mathrm{C}_{2} \mathrm{H}_{5} \mathrm{O}^{-}\) anion. Arrhenius could not tell you what is going on, but Bronsted and Lowry would have no trouble. How would they explain this reaction?

Are a \(1 \mathrm{M}\) solution of ammonia, \(\mathrm{NH}_{3}\), and a \(1 \mathrm{M}\) solution of lithium hydroxide, I iOH, equally basic, or is one solution more basic than the other? Explain.

Consider the autoionization of water. (a) What do we mean by autoionization of water (b) Write a balanced equation to go along with your explanation. (c) An alternate name for autoionization of water is autodissociation. Exactly how is water dissociating?

What is the \(\mathrm{pH}\) of a solution whose hydronium ion concentration is \(0.0010 \mathrm{M?}\) Is the solution acidic or basic?
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