Problem 26 - Introductory Chemistry Atoms First

Question

What is the logarithm of \(1.0 \times 10^{-11}\) and of \(10^{-11} ?\)

Step-by-Step Solution

Answer

The logarithms of \(1.0 \times 10^{-11}\) and \(10^{-11}\) are both -11.

Step 1: Find log of \(1.0 \times 10^{-11}\)

Using the properties of logarithms, we can rewrite the given expression: \[\log(1.0 \times 10^{-11}) = \log(1.0) + \log(10^{-11})\] Now, we know that \(\log(1) = 0\), and for the base-10 logarithm, \(\log(10^x) = x\). Therefore, the expression becomes: \[0 + \log(10^{-11}) = 0 -11 = -11\] The logarithm of \(1.0 \times 10^{-11}\) is -11.

Step 2: Find log of \(10^{-11}\)

Since we are using base-10 logarithm, we know that \[\log(10^x) = x\] So, the logarithm of \(10^{-11}\) is -11. Therefore, the logarithms of \(1.0 \times 10^{-11}\) and \(10^{-11}\) are both -11.

Key Concepts

Properties of Logarithms

Logarithms have special properties that make them useful for simplifying complex problems. The main properties include product, quotient, and power rules, which help in manipulating logarithmic expressions.

• The Product Rule states that the logarithm of a product is the sum of the logarithms: \( \log_b(MN) = \log_b(M) + \log_b(N) \).
• The Quotient Rule states that the logarithm of a quotient is the difference of the logarithms: \( \log_b\left(\frac{M}{N}\right) = \log_b(M) - \log_b(N) \).
• The Power Rule states that the logarithm of a power is the exponent times the logarithm of the base: \( \log_b(M^n) = n \cdot \log_b(M) \).

These properties help break down problems into simpler pieces. In the given exercise, the Product Rule assists in separating the expressions to calculate the logarithm of each component easily.

Base-10 Logarithm

The base-10 logarithm, also known as the common logarithm, is denoted as \( \log \) when no base is specified. This type of logarithm answers the question: "What power must 10 be raised to, to equal this number?"

For example, \( \log(1000) = 3 \) because 10 raised to the power of 3 equals 1000. The common logarithm simplifies many calculations, especially in scientific contexts, because it aligns with our base-10 number system.

In the exercise provided, the calculation of \( \log(10^{-11}) \) using base-10 logarithm is straightforward: it results directly in the exponent, which is -11. This property makes base-10 logarithms particularly handy for evaluating expressions involving powers of ten.

Scientific Notation

Scientific notation is a way to express very large or very small numbers in a compact form. It involves writing numbers as a product of a decimal and an exponent of 10. This is particularly useful in fields such as science and engineering.

For instance, the number 0.00000000001 can be expressed in scientific notation as \( 1.0 \times 10^{-11} \). Here, \( 1.0 \times 10^{-11} \) is more succinct and eliminates ambiguity.

In the context of the exercise, scientific notation helps to clearly convey the magnitude of numbers we are dealing with, allowing for an easy application of logarithmic properties to compute its logarithm.

Negative Exponents

Negative exponents offer a way to represent small numbers. This is because a negative exponent indicates the reciprocal of the base raised to the positive of that power. For example, \( 10^{-3} = \frac{1}{10^3} = 0.001 \).

When calculating logarithms, a negative exponent in the number corresponds to a negative result in the logarithm. This is seen in the calculation of \( \log(10^{-11}) \) which equals -11, indicating the power needed to scale \( 10 \) down to no more than 1 trillionth of its original size.

Understanding negative exponents is essential for interpreting results accurately, especially in logarithmic calculations where they describe the relationship between tiny values and powers of ten.