Evaluate Expressions: Unlocking the Secrets of Logarithms
Logarithms, a powerful tool in mathematics, can simplify complex expressions and solve realworld 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 stepbystep 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.
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
log_{3}27
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 3^{3} = 27, we have:
x = 3
Therefore, log_{3}27 = 3.
log_{12}121
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 12^{2} = 144 and 12^{3} = 1728, we conclude that:
121 is not a power of 12
Therefore, log_{12}121 does not exist.
log_{5}1
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 5^{0} = 1, we have:
x = 0
Therefore, log_{5}1 = 0.
log_{2}128
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 2^{7} = 128, we have:
x = 7
Therefore, log_{2}128 = 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)

What is the base of a logarithm?
A logarithm has a positive number not equal to 1 as its base. 
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. 
What is the value of log_{10}100?
log_{10}100 = 2 
Can all logarithmic expressions be evaluated?
Not all logarithmic expressions can be evaluated. For example, log_{5}3 does not exist because 5 cannot be raised to any power to obtain 3. 
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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