Problem 30 - Algebra 2

Question

Use the Change of Base Formula to evaluate each expression. Then convert it to a logarithm in base \(8 .\) $$ \log _{2} 7 $$

Step-by-Step Solution

Answer

\(\frac{\log_{8}7}{\log_{8}2} \)

Step 1: Understand the Problem

The problem requires converting the base of the logarithm from 2 to 8. To do this, we first apply the change of base formula.

Step 2: Apply the Change of Base Formula

By applying the change of base formula, we convert \(\log_{2}7\) to \(\frac{\log 7}{\log 2}\). This is because the formula allows us to write any logarithm \(\log_{b} a\) as \(\frac{\log a}{\log b}\) .

Step 3: Convert to Base 8

Now we take the result from Step 2 and convert it to base 8. Using the change of base formula again, \(\frac{\log 7}{\log 2}\) becomes \(\frac{\log_{8}7}{\log_{8}2}\).

Key Concepts

Logarithmic Expressions

Logarithmic expressions are vital in mathematics because they allow us to work with very large or very small numbers more easily. They are expressions that involve logarithms, which answer the question, "To what exponent must a base be raised to produce a given number?" For example, in the expression \(\log_b{a}\), \({b}\) is the base and \({a}\) is the number we want to express as a power of \({b}\).

Logarithms use base 10 as a common base, called the common logarithm, denoted by \(\log\). Another frequently used base is 2, which is the binary logarithm, denoted by \(\log_2\), often used in computing. There are also natural logarithms that have a base \(e\), an irrational constant roughly equal to 2.718, represented by \(\ln\).

To solve problems involving logarithmic expressions efficiently, understanding and applying laws of logarithms can simplify the computations greatly. These include the product rule, the quotient rule, and the power rule, among others.

Base Conversion

In the world of logarithms, converting a logarithm to a different base can simplify calculations and comparisons. The change of base formula is the key tool for this conversion.

Understanding the Change of Base Formula

The change of base formula states that you can convert a logarithm with any base into a fraction of two different logarithms with a common base. The formula is as follows:

• \[ \log_{b}{a} = \frac{\log_{k} a}{\log_{k} b} \]

Here, \(b\) is the original base, \(a\) is the number, and \(k\) is the new base you’re converting to. Most often, base 10 or base \(e\) is used for simplicity and availability on calculators.

Listening to the Change of Base Formula in Action

Take the example from the exercise, where you need to convert \(\log_2 7\) using the currency of base 8 via the change of base formula:

• First, use the formula with a common logarithmic base, such as 10: \[ \log_2 7 = \frac{\log 7}{\log 2} \]
• Next, convert to base 8 using the change of base formula: \[ \frac{\log_{8}7}{\log_{8}2} \]

These steps illustrate how simple it is to change the base of a logarithm and how that can assist you in evaluating or comparing logarithms across common bases.

Logarithms

Logarithms are an essential concept in algebra and many practical applications in science and engineering. They are the inverse operations of exponentiation, meaning they "undo" the process of raising a number to a power.

The Basics of How Logarithms Work

A logarithm answers the question, "How many of one number, called the base, does it take to make another number?" If you have an expression like \(b^x = a\), then the logarithm \(\log_b a = x\). This relationship helps in solving exponential equations and makes complex multiplication problems manageable.

Why Use Logarithms?

Logarithms allow us to work with large numerical data in a more accessible way. By transforming multiplicative relationships into additive ones, logarithms simplify calculations. This is particularly useful in fields like data science, physics, and financial calculations.

Moreover, learning logarithms assists in understanding phenomena that have exponential growth or decay patterns, such as population growth and radioactive decay. Recognizing and calculating with logarithms provides a deeper insight into various practical and theoretical contexts.