Problem 106 - College Algebra

Question

Evaluate each expression without using a calculator. $$ \log _{5}\left(\log _{2} 32\right) $$

Step-by-Step Solution

Answer

The result of the expression \(\log _{5}\left(\log _{2} 32\right) = 1\).

Step 1: Evaluate the inner logarithm

The logarithm \(\log_{2}32\) asks what power we must raise 2 to in order to get 32. As \(2^5 = 32\), this means that \(\log_{2}32 = 5\).

Step 2: Substitute the result into the outer logarithm

Substitute the result from the first step into the outer logarithm. This gives \(\log_{5}5\).

Step 3: Evaluate the outer logarithm

The logarithm \(\log_{5}5\) asks what power we must raise 5 to in order to get 5. As \(5^1 = 5\), this means that \(\log_{5}5 = 1\) .

Key Concepts

Evaluating Logarithms

Evaluating logarithms is like solving a mystery. You are figuring out what power a base must be raised to, in order to achieve a certain number.

To simplify a logarithmic expression, you break it down step by step.

• Start by identifying the base of your logarithm, the number you want to raise.
• Next, determine the number you want to reach by raising your base. Your goal is to find the power or exponent that balances the equation.
• Finally, use your knowledge of exponents to link the base and the desired number.

In our original exercise, we first evaluated the inner part, \(\log_{2}32\), asking 'What power must 2 be raised to, to get 32?' Recognizing that \(2^5 = 32\), we solved it: \(\log_{2}32 = 5\). Each step brings you closer to solving the mystery.

Properties of Logarithms

Understanding the properties of logarithms is key to simplifying them. These properties are rules that help us manipulate and understand logarithmic expressions.One fundamental property is that \(\log_{b}b = 1\), called the "Basic Logarithm Property." It tells us any logarithm with identical base and argument is always 1 since any number raised to the power of 1 is itself.Another property is the "Product Rule," which states \(\log_{b}(mn) = \log_{b}(m) + \log_{b}(n)\). When you multiply inside the log, you can split it into addition.

There's also the "Quotient Rule," where division inside a log results in subtraction: \(\log_{b}(\frac{m}{n}) = \log_{b}(m) - \log_{b}(n)\).

• Product Rule
• Quotient Rule
• Power Rule: \(\log_{b}(m^n) = n \cdot \log_{b}(m) \)

Understanding these properties helps simplify complex logarithmic expressions, making them easier to evaluate. In our exercise, using \(\log_{5}5\), we see the Basic Property in action, which directly tells us the result is 1.

Logarithmic Expressions

A logarithmic expression is a way of representing repeated multiplication in a digestible format. They are crucial elements of mathematics, especially useful for managing very large or small numbers by compressing data into compact forms.A logarithmic expression consists of three parts: the base, the argument, and the logarithm itself, forming something like \(\log_{b}(x)\), where 'b' is your base, and 'x' is your argument. It's asking, "To what power do we raise 'b' to get 'x'?"

Logarithmic expressions often simplify complex calculations:

• They turn multiplicative operations into additive ones, thanks to their properties.
• They allow easier comparison of numbers with vastly different magnitudes.
• They are commonly used in scientific fields such as astronomy and biology.

In our initial problem, both the inner and outer parts involve logs. The expression \(\log _{5}\left(\log_{2} 32\right)\) is simplified by evaluating the inner expression first. By consistently applying the foundational concept of logs, you can reduce the expression step by step and find clarity in your calculations.