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Chapter 15: Problem 136

What is a base-10 logarithm?

Short Answer

Expert verified
A base-10 logarithm, also known as the common logarithm, is the inverse operation to exponentiation with base 10. It helps us find the exponent to which 10 has to be raised to obtain a given value. It is represented as \(\log_{10} b\) or simply as \(\log b\). For example, \(\log_{10} 1000 = 3\) or \(\log 1000 = 3\) since \(10^3 = 1000\).

Step by step solution

01

Definition of Logarithm

A logarithm is the inverse operation to exponentiation. It helps us find the exponent to which a specific base has to be raised to obtain a given value. In general form, if \(a^x = b\), then the logarithm of b with base a is \(x\), denoted as \(\log_a b = x\).
02

Base-10 Logarithm (Common Logarithm)

A base-10 logarithm, also known as the common logarithm, uses 10 as its base. It means that we are looking for the exponent to which 10 has to be raised to get the given number. The base-10 logarithm is represented as \(\log_{10} b\) or simply as \(\log b\).
03

Practical Example

Let's consider the number 1000. To find the base-10 logarithm of 1000, we need to figure out the exponent to which 10 needs to be raised to result in 1000. Since \(10^3 = 1000\), the base-10 logarithm of 1000 is 3, represented as \(\log_{10} 1000 = 3\) or \(\log 1000 = 3\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Logarithm Definition
Understanding the concept of logarithms is fundamental in various fields of mathematics and science. A logarithm answers the question: To what exponent must we raise a certain base number to get another number? Essentially, it's a way to solve for an unknown exponent in exponential equations.

For instance, consider the equation where a raised to the power of x equals b, written as \(a^x = b\). The logarithm of b with respect to base a, symbolized as \(\log_a b\), is the exponent x. So, when we write \(\log_a b = x\), we're simply finding the exponent x that makes \(a^x = b\) true. This operation is the bedrock of what logarithms are all about.
Exponentiation
Exponentiation is a mathematical operation where a number, known as the base, is raised to a power, which is the exponent. This process multiplies the base by itself as many times as indicated by the exponent.

For example, the expression \(2^3\) denotes 2 raised to the power of 3, which equals 2 × 2 × 2, or 8. Exponentiation is the counter operation to logarithms. While exponentiation helps find the total after repeatedly multiplying a number, logarithms help us backtrack to find the original exponent, embodying the concept of inverse operations.
Common Logarithm
The base-10 logarithm, commonly referred to as the common logarithm, signifies logarithms that use 10 as their base. The symbol \(\log b\) without a base usually implies that the base is 10, as in \(\log_{10} b\).

This particular logarithm is extensively used because our number system is decimal (base-10), which makes calculations with common logarithms intuitively easier to grasp and work with. In computing the common logarithm, you're essentially asking, 'To what power must 10 be raised to attain a specific number?' For example, the common logarithm of 1000 is 3 since \(10^3 = 1000\). This is denoted as \(\log 1000 = 3\).
Inverse Operations in Mathematics
Inverse operations are pairs of mathematical operations that reverse the effects of each other, such as addition and subtraction or multiplication and division. Logarithms and exponentiation are another pair of such inverse operations.

Understanding the relationship between a function and its inverse is crucial, as it enables the solving of equations by applying the inverse operation. For example, if you're given \(2^x = 8\), you can use the inverse operation, the logarithm, to solve for x, finding that \(x = \log_2 8\), which equals 3. This illustrates how logarithms serve as a bridge to simplify and solve exponential equations.

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Most popular questions from this chapter

When \(\mathrm{Fe}^{3+}\) ions dissolve in water, they immediately form \(\left[\mathrm{Fe}\left(\mathrm{OH}_{2}\right)_{6}\right]^{3+}\) aqueous ions in which six molecules of water bind to the iron via six Fe-O bonds. However, this is not all that happens. The solution also becomes acidic. Interestingly, \(\mathrm{Fe}^{2+}\) ions also form similar \(\left[\mathrm{Fe}\left(\mathrm{OH}_{2}\right)_{6}\right]^{2+}\) ions, but the solution does not become acidic. Explain why one solution becomes acidic and the other does not. (Hint: The more positive the metal center, the more it attracts electrons to itself. Think about the effect this has on bonds within the ions.)

A solution is formed by mixing \(50.0 \mathrm{~mL}\) of \(0.015 \mathrm{M} \mathrm{NaOH}(a q)\) with \(50.0 \mathrm{~mL}\) of \(0.010 \mathrm{M}\) \(\mathrm{HNO}_{3}(a q)\) (a) Is the solution acidic or basic? (b) Draw a beaker and show all species present in the solution. (c) What is the pH of the solution?

In each of the following pairs, which is the stronger acid? (a) \(\mathrm{HPO}_{4}^{2-}\) and \(\mathrm{H}_{2} \mathrm{PO}_{4}^{-}\) (b) \(\mathrm{H}_{2} \mathrm{O}\) and \(\mathrm{H}_{3} \mathrm{O}^{+}\) (c) \(\mathrm{HCN}\left(K_{\mathrm{eq}}=6.2 \times 10^{-10}\right)\) and \(\mathrm{HCO}_{2} \mathrm{H}\left(K_{\mathrm{eq}}=1.8 \times 10^{-4}\right)\) (d) HI and HF

To be a weak base in water, a molecular compound must also be a weak electrolyte. What must be one of the ions it produces in water?

Solid ammonium chloride, \(\mathrm{NH}_{4} \mathrm{Cl}\), reacts with solid sodium hydroxide to produce ammonia gas, water, and sodium chloride, \(\mathrm{NaCl}\). (a) Write a balanced equation for this reaction. (b) According to the Bronsted-Lowry definition, which species is the acid and which species is the base? Explain.
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