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Chapter 3: Problem 38

Evaluate each expression without using a calculator. $$\log _{6} 1$$

Short Answer

Expert verified
The evaluation of the expression \(\log_6 1 = 0\).

Step by step solution

01

Understand the logarithmic function

A logarithmic function is the inverse of an exponential function. In a simple term, the logarithm of a given number b to base a is the exponent to which we have to raise a to obtain b.
02

Apply the rules of logarithmic functions

The rule which is applied here is: the logarithm of 1 to any base is always 0. Any number raised to the power of 0 is 1.
03

Evaluation of the expression \(\log_6 1\)

Based on the rule stated above, \(\log_6 1 = 0\). So, the evaluation of the given expression \(\log_6 1\) equals to 0.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Evaluating Logarithms
Understanding how to evaluate logarithms is essential in mathematics, particularly in algebra and calculus. The logarithm of a number is, fundamentally, the answer to the question: 'To what power should the base be raised, to yield this number?'
For example, when you see \( \log_b(a) = c \), you're looking at a logarithmic expression where \( b \) is the base, \( a \) the number and \( c \) the exponent or power. To evaluate a logarithm such as \( \log_6 1 \), ask yourself '6 to what power equals 1?' The answer, based on the definition of exponents, is 0, because any non-zero number raised to the power of 0 equals 1.
Inverse of Exponential Functions
Exponential functions are one-to-one functions, which means that their inverses exist. The logarithmic function is that very inverse. An exponential function, for instance, can be represented as \( y = b^x \), while its inverse logarithmic function will be \( x = \log_b(y) \). The two functions are mirror images across the line \( y = x \).
Understanding this inverse relationship is crucial in solving logarithmic equations, since it allows you to switch between expressions of exponential functions and logarithms, depending on what is required for the problem you're solving.
Logarithm Rules
There are certain rules that simplify evaluating logarithmic expressions. Some of the fundamental ones include:
  • The Product Rule: \( \log_b(mn) = \log_b(m) + \log_b(n) \)
  • The Quotient Rule: \( \log_b(\frac{m}{n}) = \log_b(m) - \log_b(n) \)
  • The Power Rule: \( \log_b(m^p) = p \cdot \log_b(m) \)
  • The Base Change Rule: \( \log_b(a) = \frac{\log_c(a)}{\log_c(b)} \)
  • The Rule of 1: \( \log_b(1) = 0 \)
  • The Rule of the Base: \( \log_b(b) = 1 \)
These rules serve as powerful tools to break down complex logarithmic expressions into more manageable pieces that can be easily evaluated.
Logarithmic Expressions
A logarithmic expression involves the log function and its variable or variables. For instance, \( \log_6 1 \) is a logarithmic expression with base 6. When working with logarithmic expressions, it's essential to identify the base and recognize that you're dealing with a function that tells you the exponent needed to reach a certain number with that base.
While evaluating logarithms, translating between exponential and logarithmic form can help to simplify the process, as different scenarios may call for different forms of an expression for an easier solution. Always remember that the base of the logarithm and the base of the corresponding exponential function are always the same.

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Most popular questions from this chapter

In Exercises \(125-132,\) use your graphing utility to graph each side of the equation in the same viewing rectangle. Then use the \(x\) -coordinate of the intersection point to find the equation's solution set. Verify this value by direct substitution into the equation. $$\log (x+3)+\log x=1$$

In Exercises \(125-132,\) use your graphing utility to graph each side of the equation in the same viewing rectangle. Then use the \(x\) -coordinate of the intersection point to find the equation's solution set. Verify this value by direct substitution into the equation. $$5^{x}=3 x+4$$

Students in a mathematics class took a final examination. They took equivalent forms of the exam in monthly intervals thereafter. The average score, \(f(t),\) for the group after \(t\) months was modeled by the human memory function \(f(t)=75-10 \log (t+1), \quad\) where \(\quad 0 \leq t \leq 12 . \quad\) Use \(\quad\) a graphing utility to graph the function. Then determine how many months elapsed before the average score fell below 65.

In Exercises \(125-132,\) use your graphing utility to graph each side of the equation in the same viewing rectangle. Then use the \(x\) -coordinate of the intersection point to find the equation's solution set. Verify this value by direct substitution into the equation. $$2^{x+1}=8$$

The formula \(S=C(1+r)^{t}\) models inflation, where \(C=\) the value today, \(r=\)the annual inflation rate, and \(S=\)the inflated value t years from now. Use this formula to solve. Round answers to the nearest dollar. If the inflation rate is \(3 \%,\) how much will a house now worth \(\$ 510,000\) be worth in 5 years?
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