Problem 27 - Introductory Chemistry Atoms First

Question

What is the negative logarithm, \(-\log\), of \(10^{-7}\) ? (Hint: The minus sign in -log means you must put a minus sign in front of the number you get for the logarithm.)

Step-by-Step Solution

Answer

The short answer is: \(-\log_{10}(10^{-7}) = 7\).

Step 1: Define the negative logarithm formula

Since we are dealing with the negative common logarithm, the formula is: \(-\log_{10}(x) = -\log_{10}(10^{-7})\)

Step 2: Apply the logarithm rule

When the base of the logarithm and the number inside the logarithm are the same, the logarithm of that number is equal to one. Therefore, we can write the logarithm equation as: \(\log_{10}(10^y) = y\)

Step 3: Find the logarithm of the given number

Since \(x = 10^{-7}\), we plug this value into the formula and solve for y: \(\log_{10}(10^{-7}) = -7\)

Step 4: Find the negative logarithm

Now that we know the logarithm is -7, we can find the negative logarithm by applying the hint provided in the question: \(-\log_{10}(10^{-7}) = -(-7) = 7\) The negative logarithm of \(10^{-7}\) is 7.

Key Concepts

Logarithm Rules

Logarithms can be tricky, but following the rules makes them easier to understand. There's a fundamental rule that says if you have a logarithm with the same base and the number inside, it's pretty straightforward.
Take for example \( \log_{b}(b^y) = y \. This means if we have logarithm base \(b\) of \(b\) raised to some power \(y\), then our answer is simply \(y\).
This clever rule is what simplifies so many complex-looking problems.
Also, if you see a minus sign in front of a logarithm (like \-\log\), it indicates you need to place that minus after solving the log, but then flip it afterwards as in the exercise example.

Common Logarithm

The term "common logarithm" refers to a logarithm that uses base 10. It is often written as \(\log_{10}\) or simply \(\log\) when the context is clear. In many cases, logarithms without any base mentioned are common logarithms.
These are incredibly useful thanks to how our number system is based on powers of 10.
Common logarithms allow you to express powers of 10 in a compact form.
For instance, the common logarithm of 10,000 is 4, because 10 raised to the power of 4 equals 10,000.

Exponentiation

Exponentiation is simply the operation of raising a number, called the base, to the power of another number. It's like repeated multiplication.
For example, when we have \(10^{-7}\), it means 10 is our base and it's raised to the power of -7.
This operation can also be seen as taking 1 over 10 raised to the positive power 7, which results in a really small decimal that looks like 0.0000001.
Understanding this process is key in logarithms since logs are like the reverse operation of exponentiation.
Where exponentiation multiplies, logarithms count how many times you multiply.

Base 10 Logarithm

A base 10 logarithm is a type of logarithm where the base number is 10. This base is commonly used due to its simplicity and because our number system is based on 10.
In science, engineering, and many branches of mathematics, base 10 logarithms are heavily used and quite practical.
When you see \(\log_{10}\), it's hinting at a relationship with powers of 10, which makes calculations more intuitive.
In the exercise \-\log_{10}(10^{-7})\, the base 10 helped simplify the problem by allowing immediate application of the logarithm rule, making it easy to derive that the negative logarithm of \(10^{-7}\) is not actually negative, but positive 7.