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Chapter 12: Problem 35

Simplify. $$ 5^{\log _{5} 7} $$

Short Answer

Expert verified
7

Step by step solution

01

Understand the given expression

The given expression is an exponential expression with a logarithm as the exponent: \( 5^{\log _{5} 7} \)
02

Identify the properties of logarithms

Recall the property of logarithms: If \( a^{\log_{a}(b)} = b \).
03

Apply the logarithmic property

Applying the property \( a^{\log_{a}(b)} = b \) to our expression: \( 5^{\log _{5} 7} = 7 \)

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Key Concepts

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

properties of logarithms
Logarithms have several properties that simplify complex expressions. One vital property to remember is that logarithms are the inverse of exponents. This means that they can 'undo' each other. The property we used in this exercise is:
If you have an expression of the form \( a^{\text{log}_{a}(b)} \), it simplifies directly to \( b \).
  • For example, \( 3^{\text{log}_{3}(81)} = 81 \)
  • This property leverages the inverse relationship between exponents and logarithms.
Remembering this type of property can help make simplifying tricky expressions a lot easier.
logarithmic and exponential relationship
Logarithms and exponents are tightly connected. They are essentially two sides of the same coin.
  • The expression \( \text{log}_a(b) \) means 'to what power must \( a \) be raised, to get \( b \)?'
  • Conversely, \( a^c = b \) where \( c = \text{log}_a(b) \).
This relationship is powerful because it allows you to switch between the two forms to simplify problems. For example:
  • If you have \( 2^{\text{log}_2(8)} \), you can use your knowledge of the logarithmic and exponential relationship to immediately know this simplifies to 8.
Grasping this relationship helps in solving exponential and logarithmic computations more efficiently.
simplification techniques
When simplifying expressions, several techniques can ensure your process stays smooth and accurate:
  • **Identify Key Properties**: Recognize properties of exponents and logarithms that can simplify your expression.
  • **Express in Simpler Forms**: Convert complex expressions into simpler forms using inverse properties.
  • **Practice Regularly**: The more you practice, the more familiar you'll become with these properties, making simplifications faster and more intuitive.
Using these strategies, let's revisit our original problem, \( 5^{\text{log}_5(7)} \):
  • Recognize the property \( a^{\text{log}_a(b)} = b \).
  • Apply this property to get \( 5^{\text{log}_5(7)} = 7 \).
These steps demonstrate the efficiency of understanding and applying simplification techniques.

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Most popular questions from this chapter

Simplify. $$ \log _{10}\left(\log _{4}\left(\log _{3} 81\right)\right) $$

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Show that for exponential growth at rate \(k,\) the doubling time \(T\) is given by \(T=\frac{\ln 2}{k}\)

Solve. $$ \log _{4}(3 x-2)=2 $$
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