Problem 40 - Algebra and Trigonometry

Question

In Exercises 21–42, evaluate each expression without using a calculator. $$ \log _{4} 4^{6} $$

Step-by-Step Solution

Answer

The value of the expression \( \log _{4} 4^{6} \) is 6.

Step 1: Understand the logarithm rule

The rule that relates to this problem states that for any base b and any exponent p, \(\log_{b}(b^{p}) = p\). Essentially, taking the base b logarithm of b raised to any power automatically simplifies to that power.

Step 2: Apply the logarithm rule

In our case, the base is 4, and the exponent is 6. So when we apply the rule, we get our result: \( \log_{4}(4^{6}) = 6 \).

Key Concepts

Logarithmic Expressions

Logarithmic expressions are a way to represent the exponent that a base number must be raised to in order to produce a given number. A logarithm asks the question, 'To what power must we raise the base to obtain a certain number?' For instance, the expression \( \log_{b}(x) \) tells us the power to which the base \( b \) must be raised to get \( x \). In the problem \( \log_{4}(4^{6}) \) the base is \( 4 \) and the expression essentially translates to 'what power do we raise 4 to get \( 4^{6} \)?' The answer to this, guided by logarithmic rules, simplifies to just the exponent, which is \( 6 \). Understanding these expressions is crucial as they are the inverse functions of exponentiation.

Knowing how to rewrite and simplify logarithmic expressions can be tremendously helpful in various areas of mathematics, including solving exponential equations, analyzing logarithmic scales, and working with growth and decay models in science and economics.

Evaluating Logarithms

Evaluating logarithms involves finding the value of a logarithmic expression without the aid of a calculator. This often requires applying specific logarithm rules that can simplify the process. One such rule is that \( \log_{b}(b^{p}) = p \), where \( b \) is the base and \( p \) is the exponent. It's important to note that if the base and the number inside the log function are the same, the result is simply the exponent. These foundational rules are key to evaluating more complex logarithmic equations as well. In the given exercise \( \log_{4}(4^{6}) \), by applying this rule, we find that the base (4) raised to the power of what number gives us \( 4^{6} \) once again? The answer is straightforward: \( 6 \).

It's vital to become comfortable with these basic evaluations before moving on to more advanced topics, such as changing the base of a logarithm or dealing with logarithms in different bases. This knowledge acts as a stepping stone to understanding logarithms in algebra, calculus, and beyond.

Exponentiation

Exponentiation is an arithmetic operation where a number, known as the base, is raised to the power of an exponent. The exponent tells us how many times the base needs to be multiplied by itself. For instance, \( 3^{4} \) means you multiply 3 by itself 4 times: \( 3 \times 3 \times 3 \times 3 \). In the context of logarithms, understanding exponentiation is imperative because a logarithm is essentially the inverse of exponentiation. When you see an expression like \( \log_{b}(x) = y \) it answers the question, 'To what exponent must the base \( b \) be raised to produce \( x \)?'

Therefore, the relationship between logarithms and exponentiation is a fundamental concept which students must master to move through topics in algebra and advanced mathematics. In our exercise, the exponentiation part is straightforward: \( 4 \) is our base, and it is being raised to the power of \( 6 \) which, when inverted through the lens of logarithms, results in a simple operation where the exponent \( 6 \) is the answer.