Problem 39 - Algebra for College Students

Question

Evaluate each logarithmic expression. $$ \log _{10}\left(\log _{7} 7\right) $$

Step-by-Step Solution

Answer

The expression evaluates to 0.

Step 1: Understanding the inner logarithm

First, focus on simplifying the inner logarithmic expression: \( \log_{7} 7 \). The logarithm \( \log_{b} b \) always equals 1 because \( b^{1} = b \). Therefore, \( \log_{7} 7 = 1 \).

Step 2: Evaluating the outer logarithm

Now, substitute the result from the inner expression into the outer logarithm: \( \log_{10}(1) \). By the definition of logarithms, \( \log_{10}(1) = 0 \) because \( 10^{0} = 1 \).

Key Concepts

Logarithmic Expressions

A logarithmic expression is essentially a mathematical way to express the concept of exponents. When you see something like \( \log_b(a) \), it's asking you: "To what power must \( b \) be raised to get \( a \)?".

Logarithms have a special property: they can transform multiplication into addition, making calculations easier. For example, if you have two numbers, \( x \) and \( y \), the expression for their logarithms is \( \log_b(xy) = \log_b(x) + \log_b(y) \). This property is particularly useful in computer science and information theory because it simplifies complex multiplication into simple addition.

When dealing with logarithmic expressions, breaking them down into smaller parts can make them more manageable. Start by checking if simplifications are possible with properties like the change of base formula or understanding the meaning of \( \log_b(b) \).

Base of Logarithm

The base of a logarithm is a crucial component that determines how the logarithm is calculated.

Let's consider a logarithm \( \log_b(n) \) where \( b \) is the base. The base tells us the number we multiply by itself to reach \( n \). Consider the logarithm \( \log_{7}(7) \), where our base is 7. In this simple case, the answer is 1 because \( 7^1 = 7 \).

The most commonly used bases are:

• Base 10, often called the "common logarithm." It's frequently used in general mathematics and science.
• Base \( e \) (where \( e \) is approximately 2.718), known as the "natural logarithm." This is commonly used in calculus and exponential growth models.

Choosing the correct base can make solving a logarithmic expression a lot easier by simplifying calculations according to the context.

Evaluate Logarithms

Evaluating logarithms involves determining the power to which the base must be raised to produce the given number. This is the crux of understanding logarithmic expressions: solving for the exponent.

For instance, take the expression \( \log_{10}(1) \). The task is to find the number \( x \) such that \( 10^x = 1 \). The only number that works here is 0 because any number raised to the power of 0 equals 1.

To evaluate more complex logarithms:

• Simplify the equation if possible, breaking it into smaller, more manageable parts.
• Identify basic properties such as \( \log_b(b) = 1 \), which can streamline your calculations.
• Apply known mathematical rules, like change of base, if necessary, to make the logarithm easier to compute.

These steps and concepts will help you accurately evaluate logarithms, making them useful tools rather than sources of confusion.