Question

Use the change-of-base formula to evaluate the logarithm. $$\log _2 28$$

Short answer

The value of \(\log_2 28\) is another way of saying \(\frac{\log_{10} 28}{\log_{10} 2}\) using the change-of-base formula.

Step-by-step solution

Identifying the variables in the formula

In the given expression \(\log_2 28\), \(b = 2\) and \(a = 28\). We are to use the change-of-base formula to find the value of the expression.

Choosing a new base

For the new base \(k\), we generally choose 10 or \(e\) since most calculators have buttons for \(\log_{10}(x)\) and \(\log_{e}(x)\). Here we choose \(k = 10\) for simplicity.

Apply the change of base formula

Now we apply the change-of-base formula which is \(\log_b a = \frac{\log_k a}{\log_k b}\). Substituting values we get \(\log_2 28 = \frac{\log_{10} 28}{\log_{10} 2}\).

Key concepts

Logarithms

Logarithms are an essential concept in mathematics that allow us to understand how numbers behave when they are multiplied or divided. Essentially, a logarithm answers the question: "To what power must a specific base be raised, to produce a given number?" For example, in the expression \( \log_b a \), \(b\) represents the base, and \(a\) is the number we are trying to find the power for. This means we are finding out what power we should raise \(b\) to, in order to get \(a\).

• A logarithm can be thought of as the opposite of exponentiation.
• It is useful in dealing with exponential growth or decay situations.
• Logarithms are vital in fields such as science, engineering, and economics due to their ability to simplify complex multiplicative processes.

Understanding the change-of-base formula becomes important when you encounter logarithms with bases that aren't easy to work with. This formula allows you to convert a logarithm with a difficult base into one that is easier to calculate, usually using base 10 or the natural base \(e\). This is particularly useful because most calculators have these base operations readily available.

Base 10 Logarithm

The base 10 logarithm, often denoted \(\log\) or \(\log_{10}\), is a logarithm where the base \(b = 10\). This is also known as the common logarithm. Base 10 is widely used because of our decimal numbering system.

• The base 10 logarithm of a number tells you how many times you multiply 10 to get that number.
• For example, \(\log_{10} 100 = 2\), because 10 raised to the power of 2 equals 100.
• Base 10 logarithms are particularly handy since many calculators feature an easy-to-use button for \(\log_{10}\).

When using the change-of-base formula, the base 10 logarithm is often chosen due to this calculator convenience. It allows for straightforward computation, turning log problems into simple division problems using \(\log_{10}\) of the numbers involved.

Base 2 Logarithm

The base 2 logarithm, often referred to as the binary logarithm and denoted by \(\log_2\), is unique compared to the common logarithm because its base is 2. This type of logarithm is particularly useful in computer science, as computers operate in binary.

• The binary logarithm of a number answers how many times you would need to multiply 2 to get that number.
• For example, \(\log_2 8 = 3\), because 2 to the power of 3 equals 8.
• In fields dealing with binary data, \(\log_2\) helps in determining efficiencies and data throughput.

Because calculators usually don’t have a direct \(\log_2\) function, the change-of-base formula becomes an essential tool. By using this formula, you can convert \(\log_2\) into \(\log_{10}\) or natural logarithms which are more easily determined with standard calculators. This conversion helps in evaluating expressions like \(\log_2 28\) efficiently.