Problem 37 - Algebra for College Students

Question

Evaluate each logarithmic expression. $$ \log _{5}\left(\log _{2} 32\right) $$

Step-by-Step Solution

Answer

The expression evaluates to 1.

Step 1: Evaluate the Inner Logarithm

First, evaluate the expression inside the parentheses: \(\log_{2} 32\). Since 32 is a power of 2 (i.e., \(32 = 2^5\)), we can determine that \(\log_{2} 32 = 5\).

Step 2: Evaluate the Outer Logarithm

Now substitute the result from the first step into the outer logarithm: \(\log_{5} 5\). Since 5 is the base of the logarithm and also the argument, \(\log_{5} 5 = 1\).

Key Concepts

Understanding Logarithmic Functions

A logarithmic function is essentially the inverse of an exponential function. When you have an equation like \( log_b(x) = y\), it means that the base \(b\) raised to the power of \(y\) equals \(x\). This translates to the expression \(b^y = x\). Logarithmic functions are useful for solving problems involving exponential growth or for simplifying complex exponential expressions.

Key Points:

• The base number is the subscript of the logarithm.
• Logs can help us find how many times one number needs to be multiplied by itself to achieve another number.
• Understanding the relationship between logarithmic and exponential expressions is crucial for problem-solving.

This concept not only helps in mathematical calculations but is extensively used in real-life applications such as computing interest rates, population growth, and more.

Evaluating Logarithms

Evaluating logarithms involves finding the exponent that the base must be raised to in order to get the expression's value. When evaluating a logarithm, it's important to recognize patterns in numbers and use the properties of logarithms efficiently. Let's go through a quick example.Imagine you're tasked with evaluating \(\log_{2} 32\). If you know that \(32 = 2^5\), it becomes obvious that \(\log_{2} 32 = 5\). Why? Because \(2\) raised to the-power-of \(5\) equals \(32\), perfectly fitting our logarithmic definition.

When evaluating really take note of:

• The properties of logarithms such as \(\log_b(b) = 1\) and \(\log_b(b^x)=x\).
• Transforming the base if possible to make computations simpler.
• Breaking the problem into smaller, more manageable parts.

This will help ensure you tackle each logarithmic expression with confidence and precision.

Mathematical Problem Solving with Logarithms

Mathematical problem-solving involving logarithms usually requires you to combine several concepts and interpret mathematical expressions accurately.

Approaching logarithmic problems:

• First, simplify the expression as much as possible. Sometimes, as in our original exercise, there are nested logarithms where solving the inner part simplifies the rest.
• Use known mathematical identities and properties to break down the problem.
• Check each step closely to avoid small mistakes, which can influence the whole solution.

As demonstrated in the example \(\log _{5}(\log _{2} 32)\), solving inner expressions first can create an easier path to the final answer. This method reduces complexity and provides clear and concise solutions, enhancing your problem-solving skills.