Problem 105 - College Algebra

Question

Evaluate each expression without using a calculator. $$ \log _{3}\left(\log _{7} 7\right) $$

Step-by-Step Solution

Answer

The value of the given logarithmic expression is 0.

Step 1: Evaluate Inner Logarithm

First, evaluate the inner logarithm, \(\log _{7} 7\). According to the definition of logarithm, since \(7^1 = 7\), we have \(\log _{7} 7 = 1\). Substitute this result into the overall expression.

Step 2: Evaluate Outer Logarithm

Next, evaluate the outer logarithm, \(\log _{3}(1)\). According to the definition of logarithm, since \(3^0 = 1\), we have \(\log _{3} 1 = 0\).

Step 3: Interpret Result

The expression \(\log _{3}\left(\log _{7} 7\right)\) has been evaluated as the number 0.

Key Concepts

Algebra

Algebra is a branch of mathematics where symbols and letters are used to represent numbers and quantities in formulas and equations. It serves as the foundational bedrock for various mathematical concepts and operations, including solving logarithmic equations like the one in our exercise.

In this problem, algebraic thinking is required to understand and perform operations with logarithms. Algebraic knowledge allows us to comprehend properties of logarithmic functions, like how they are the inverse of exponential functions, and how to simplify expressions involving them. We applied these principles to evaluate the expression by decomposing it into more manageable parts, starting with the innermost logarithm and working our way out.

Logarithmic Identity

Logarithmic identities are fundamental tools that simplify logarithmic expressions and enable us to compute them without the aid of a calculator. One key identity is that the logarithm of a number in the same base equals one: \( \log_b{b} = 1 \).

Our exercise involved \( \log_{7}{7} \) as the inner expression, which applies this identity. Understanding that any base raised to the power of zero equals one, as in \( b^0 = 1 \), we can also solve the outer logarithm \( \log_{3}{1} = 0 \) confidently. Realizing these identities are crucial in simplifying and evaluating logarithmic expressions accurately, as we saw in our step-by-step solution.

Exponential Functions

Exponential functions are mathematical functions of the form \( f(x) = a^{x} \), where the base \( a \) is a constant and \( x \) is the variable exponent. They feature prominently in the inverse operation of taking logarithms, as an integral part of understanding how logarithms work.

In our example, recognizing that the logarithm is the inverse operation of exponentiation helped to decipher the given expression. It tells us the exponent that the base of the logarithm must be raised to in order to produce a given number. Specifically, the solution involves the idea that \( \log_b{a} \) answers the question, 'To what power must \( b \) be raised, to produce \( a \)?' Applying this concept to \( \log_{3}{1} \) implies finding the power that 3 must be raised to yield 1, which is 0, as anything raised to the power of zero is 1. This concept is fundamental to not just understanding our initial problem, but also to grasping a wider range of algebraic problems involving exponential relationships.