Question

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

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

Step-by-step solution

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.

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.

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.

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.

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.

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.

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.

Key concepts

Base-10 Logarithm

The base-10 logarithm, commonly denoted as \(\log_{10} x\), is a logarithm where the base is 10. It is the power to which the number 10 must be raised to obtain a certain value. Understanding the base-10 logarithm is crucial for simplifying large numbers and solving exponential equations. For example, if \(10^y = x\), then \(\log_{10} (x) = y\).
This concept is often used in scientific fields where it is necessary to manage a wide range of values, like measuring pH in chemistry or decibels in acoustics. The base-10 logarithm transforms multiplicative processes into additive ones, making it easier to handle exponential relationships.
Let's take the example of \(\log_{10} (60)\). We are trying to find the exponent such that when 10 is raised to this power, the result is 60. By estimating within intervals, an educational technique as seen in the step-by-step solution, we can approximate \(\log_{10} (60)\) to be between 1 and 2, since 60 is more closely aligned with 100 than with 10.

Exponential Functions

Exponential functions involve constant bases raised to variable exponents, expressed generally as \(f(x) = a^x\), where \(a\) is a constant. These functions are characterized by their rapid growth or decay, depending on the base value. In specific, the base-10 function \(10^x\) is fundamental in logarithms and often appears in real-world data that involves powers of ten.
Exponential functions can be graphed as curves that increase exponentially with larger exponents or decrease exponentially with negative exponents. For instance, in our problem, the expressions \(10^{-2}\), \(10^{-1}\), and \(10^0\) signify rapid decrease as the exponents become more negative. Conversely, \(10^1\) and \(10^2\) showcase an increase.
When solving logarithm problems—like finding \(\log_{10} (60)\)—we essentially reverse the process of exponentiation. By determining between which whole power of ten the number 60 falls, we can infer the logarithmic value given the pattern of exponential growth.

Logarithmic Properties

Logarithms come with a set of intrinsic properties that simplify complex calculations and relationships. They allow us to transition from multiplicative to additive problems, an advantage when working with large numbers. Some key properties include:

• \(\log_{10} (ab) = \log_{10} (a) + \log_{10} (b)\): The logarithm of a product is the sum of the logarithms.
• \(\log_{10} (a^b) = b\log_{10} (a)\): The logarithm of a power involves multiplying the exponent by the logarithm.
• \(\log_{10} \left(\frac{a}{b}\right) = \log_{10} (a) - \log_{10} (b)\): The logarithm of a quotient is the difference of the logarithms.

By applying these properties, one can easily manage and solve exponential problems, even when they become complicated. For the problem \(\log_{10} (60)\), extrapolating interval estimation allowed us to decide that its logarithm resides between the powers 1 and 2, showcasing the additive rule: since 60 is between 10 (\(10^1\)) and 100 (\(10^2\)), we find that \(\log_{10}(60)\) is closer to 1.8 than 1.3, for instance.