Evaluate Each Expression Log327 Log121 Log5 1 25 Log2128

Evaluate Expressions: Unlocking the Secrets of Logarithms

Logarithms, a powerful tool in mathematics, can simplify complex expressions and solve real-world problems. In this guide, we will embark on a journey to evaluate five logarithmic expressions: log327, log121, log5 1, 25, and log2128.

Navigating the Challenges

Logarithms can present challenges, especially when dealing with various bases and arguments. It’s easy to get lost in the intricacies of logarithmic rules, but we will break down each expression step-by-step to make the process more accessible.

Revealing the Solutions

  • log327 = 3 (since 3^3 = 27)
  • log121 = 2 (since 11^2 = 121)
  • log5 1 = 0 (since 5^0 = 1)
  • 25 = 2 (since 2^2 = 25)
  • log2128 = 5 (since 2^5 = 32)

Key Points

  • Understanding logarithmic basics is crucial for evaluating expressions.
  • Breaking down complex expressions into simpler ones simplifies the evaluation process.
  • Applying logarithmic rules ensures accuracy in calculations.
  • The evaluated expressions provide valuable insights into the relationships between numbers and exponents.
Evaluate Each Expression Log327 Log121 Log5 1 25 Log2128

Evaluating Logarithmic Expressions

Understanding Logarithms

In mathematics, logarithms are used to represent the power to which a base must be raised to produce a given number. The base of a logarithm is always a positive number not equal to 1.

Evaluating Logarithmic Expressions

To evaluate a logarithmic expression, we need to determine the exponent (power) to which the base must be raised to obtain the argument (number). The general form of a logarithmic expression is:

log<sub>base</sub>(argument) = exponent

log327

Solution:

We need to find the exponent to which 3 must be raised to obtain 27.

log<sub>3</sub>27 = x
3<sup>x</sup> = 27

Since 33 = 27, we have:

x = 3

Therefore, log327 = 3.

log12121

Solution:

We need to find the exponent to which 12 must be raised to obtain 121.

log<sub>12</sub>121 = x
12<sup>x</sup> = 121

Since 122 = 144 and 123 = 1728, we conclude that:

121 is not a power of 12

Therefore, log12121 does not exist.

log51

Solution:

We need to find the exponent to which 5 must be raised to obtain 1.

log<sub>5</sub>1 = x
5<sup>x</sup> = 1

Since 50 = 1, we have:

x = 0

Therefore, log51 = 0.

log2128

Solution:

We need to find the exponent to which 2 must be raised to obtain 128.

log<sub>2</sub>128 = x
2<sup>x</sup> = 128

Since 27 = 128, we have:

x = 7

Therefore, log2128 = 7.

Conclusion

Logarithmic expressions are useful for solving various mathematical problems and applications. By understanding the concept of logarithms, we can evaluate logarithmic expressions accurately and efficiently.

Frequently Asked Questions (FAQs)

  1. What is the base of a logarithm?
    A logarithm has a positive number not equal to 1 as its base.

  2. How do I evaluate a logarithmic expression?
    To evaluate a logarithmic expression, find the exponent to which the base must be raised to obtain the argument.

  3. What is the value of log10100?
    log10100 = 2

  4. Can all logarithmic expressions be evaluated?
    Not all logarithmic expressions can be evaluated. For example, log53 does not exist because 5 cannot be raised to any power to obtain 3.

  5. What are the applications of logarithms?
    Logarithms are used in various fields, including mathematics, physics, chemistry, and economics, to solve problems involving exponential functions and growth rates.

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