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Chapter 5: Problem 21

Find each of the following. Do not use a calculator. $$\log _{5} 5^{4}$$

Short Answer

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Step by step solution

01

Identify the Logarithmic Property

Recognize that the expression \(\textup{log}_{5} 5^{4}\) can be simplified using the logarithmic property \(\textup{log}_{b} b^{x} = x\). This property states that the logarithm of a base raised to an exponent is equal to the exponent.
02

Apply the Property

According to the property mentioned, we can directly write the expression \(\textup{log}_{5} 5^{4}\) as the exponent, which is 4.

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Key Concepts

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

Logarithmic Properties
In mathematics, logarithmic properties help simplify equations involving exponents. One of the key properties is the Power Rule: \(\textup{log}_{b} b^{x} = x\). This means that if you take the logarithm of a base raised to an exponent, it equals that exponent.
For example, in our exercise\(\textup{log}_{5} 5^{4}\), we recognize that 5 is both the base of the logarithm and the base in the exponent, making it a perfect candidate for the Power Rule. By identifying this property, we can simplify the expression easily without complex calculations.
Exponents
Exponents represent repeated multiplication of a number by itself. For example, \(5^{4} = 5 \times 5 \times 5 \times 5\). Understanding exponents is crucial when dealing with logarithms since logarithms are essentially the inverse operations of exponents.
When solving logarithmic expressions, identifying the base and the exponent is essential. In the case of \(\textup{log}_{5} 5^{4}\), recognizing that we have an exponent of 4 helps us apply the logarithmic property correctly.
Simplification
Simplification is the process of making an expression easier to work with. Using properties and rules can make this process straightforward. With logarithms, properties like the Power Rule can greatly reduce the complexity of expressions.
Applying the Power Rule to \( \textup{log}_{5} 5^{4} \) simplifies directly to 4 because \( \textup{log}_{5} b^{x} = x \). This is a much more manageable and understandable form.
  • Identify components: Recognize parts of the expression like the base and exponents.
  • Apply properties: Use logarithmic properties relevant to the problem.
  • Simplify: Perform the indicated operations to reach the final, simplified answer.
Through simplification, we turn complicated expressions into easy-to-understand results, making mathematics both approachable and enjoyable.

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Most popular questions from this chapter

Determine whether each of the following is true or false. Assume that \(a, x, M,\) and \(N\) are positive. $$\log _{a} x^{3}=3 \log _{a} x$$

Solve. $$2 \log 50=3 \log 25+\log (x-2)$$

Solve the logarithmic equation algebraically. Then check using a graphing calculator. $$\log (2 x+1)-\log (x-2)=1$$

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