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Chapter 6: Problem 14

Rewrite the equation in logarithmic form. (See Example 2.) \(5^{-2}=\frac{1}{25}\)

Short Answer

Expert verified
The logarithmic form of the equation \(5^{-2}=\frac{1}{25}\) is \(\log_5 \left(\frac{1}{25}\right) = -2\)

Step by step solution

01

Identify the base, exponent and result in the equation

In this problem we have the base \(b = 5\), the exponent \(y = -2\) and the result \(x = \frac{1}{25}\).
02

Transformation into Logarithmic form

From Step 1, we apply the transformation of the formula \(\log_b(x) = y\). Here base is \(5\), \(x\) is \(\frac{1}{25}\) and \(y\) is \(-2\).
03

Write the Logarithmic Form

Applying our variables to the formula we get, \(\log_5 \left(\frac{1}{25}\right) = -2\) which is the logarithmic form of our original equation \(5^{-2}=\frac{1}{25}\).

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Key Concepts

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

Understanding the Base
In mathematics, when working with exponents and logarithms, the "base" is the number that is going to be raised to a power. In our example equation, the base is 5. This means when 5 is raised to an exponent (or power), it produces another number that we refer to as the result.
  • The base of an expression tells you the repeating factor.
  • In the equation \(5^{-2} = \frac{1}{25}\), 5 is the base.
Understanding the base is crucial because it represents the root number in exponential and logarithmic calculations. Keep in mind, when you express a logarithm like \(\log_b(x)\), the "b" represents the base, which stays constant through transformations from exponential to logarithmic form.
Grasping the Exponent
The exponent in an expression denotes how many times the base is used as a factor. It tells you how many times to multiply the base by itself. Negative exponents, like the one in our example \(-2\), have a special meaning鈥攖hey indicate division as opposed to multiplication.
  • The exponent \(-2\) in \(5^{-2}\) means that 5 is to be divided twice, resulting in a fraction.
  • Using negative exponents: \(a^{-n} = \frac{1}{a^n}\).
In our original equation \(5^{-2} = \frac{1}{25}\), a negative exponent shows inversion and repetition. Understanding this concept is essential as it simplifies the process of rearranging expressions into logarithmic form.
What is Logarithm Transformation
Logarithm transformation is the method of converting an exponential equation into a logarithmic equation. This change is helpful in simplifying the understanding of the equation and can be useful in solving complex mathematical problems. The transformation relies on the relationship: \( \log_b(x) = y\)
  • "b" in this expression is the base of both the logarithmic and original exponential equations.
  • "x" is the result you obtain when the base is raised to its related exponent "y".
In our exercise, we transform the exponential equation \(5^{-2} = \frac{1}{25}\) to its logarithmic form using the elements we identified:
  • Base \(b = 5\)
  • Exponent \(y = -2\)
  • Result \(x = \frac{1}{25}\)
Thus, the logarithmic form becomes \(\log_5 \left(\frac{1}{25}\right) = -2\). This transformation is a powerful tool to solve or rearrange equations in mathematics.

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