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Chapter 3: Problem 24

Use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. $$\log _{5}\left(\frac{\sqrt{x}}{25}\right)$$

Short Answer

Expert verified
\( \frac{1}{2} \log_{5}x - 2 \)

Step by step solution

01

Apply the Quotient Rule

The quotient rule states that the logarithm of a quotient is equal to the difference of the logarithms. Therefore, we can write \( \log_{5}(\frac{\sqrt{x}}{25}) \) as \( \log_{5}(\sqrt{x}) - \log_{5}(25) \)
02

Evaluate the Logarithm

The second part of our expression, \( \log_{5}(25) \), can be evaluated directly as the base 5 raised to what power gives us 25? The answer is 2. So this part of the expression simplifies to 2.
03

Apply the Power Rule

We cannot directly evaluate the first part of the expression \( \log_{5}(\sqrt{x}) \) because it depends on x. However we can simplify the square root of x as \( x^{1/2} \). Using the power rule of logarithms, we can bring the exponent in front of the logarithm. So, the \( \log_{5}(\sqrt{x}) \) becomes \( \frac{1}{2}(\log_{5}x) \)
04

Combine All Parts

Now we can combine all simplifications. Instead of the initial expression, we have \( \frac{1}{2}(\log_{5}x) - 2 \)

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Key Concepts

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

Understanding the Quotient Rule for Logarithms
To understand the quotient rule for logarithms, think of how dividing numbers works under logarithms. When you have a logarithm of a fraction, the rule helps break it down into two simpler parts. This rule tells us that the log of a quotient is the log of the numerator minus the log of the denominator.
For instance, in the expression \(\log_{5}\left(\frac{\sqrt{x}}{25}\right)\), we can apply the quotient rule.
  • The numerator is \(\sqrt{x}\), and
  • the denominator is \(25\).
Using the quotient rule, it becomes \(\log_{5}(\sqrt{x}) - \log_{5}(25)\). This simplification makes it easier to handle complex fractions, paving the way for further simplifications or evaluations.
Exploring the Power Rule for Logarithms
The power rule for logarithms allows us to simplify any logarithm that involves an exponent. When you encounter a logarithm with an exponent, this rule states that you can move the exponent in front of the logarithm as a multiplier.
In the problem we have, \(\log_{5}(\sqrt{x})\), we recognize the square root as an expression with an exponent. Recall that \(\sqrt{x}\) is the same as \(x^{1/2}\).
  • Thanks to the power rule, it transforms to \(\frac{1}{2} \log_{5}(x)\).
The power rule is particularly handy when dealing with roots or integers raised to a power. By converting a computation involving an exponent into a simple multiplication, the power rule makes solving logarithmic expressions both easier and more approachable.
Mastering Logarithm Evaluation
Logarithm evaluation helps us solve logarithms directly, without a calculator. It requires understanding what logarithms mean. When you evaluate \(\log_{5}(25)\), you're asking: What power must you raise 5 to get 25?
  • The answer is readily apparent: \(5^2 = 25\). So, \(\log_{5}(25) = 2\).
Evaluating logarithms efficiently requires familiarity with the basics of exponents. This foundational knowledge allows you to solve or simplify logarithmic expressions quickly. The better you understand how numbers interrelate through multiplication and division, the more intuitive logarithm evaluation will become.

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Most popular questions from this chapter

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