Problem 4 - Algebra A Combined Function

Question

Write each as an exponential equation. $$ \log _{5} \frac{1}{25}=-2 $$

Step-by-Step Solution

Answer

The exponential equation is \( 5^{-2} = \frac{1}{25} \).

Step 1: Understand what the logarithm tells us

The problem given is \( \log_{5} \frac{1}{25} = -2 \), which means that 5 raised to the power of -2 gives \( \frac{1}{25} \). Thus, we need to convert this logarithmic form into its equivalent exponential form.

Step 2: Convert the logarithmic equation to exponential form

To write the logarithm \( \log_{5} \frac{1}{25} = -2 \) as an exponential equation, we use the definition of logarithms: If \( \log_{b} a = c \), then \( b^{c} = a \). Here, \( b = 5 \), \( c = -2 \), and \( a = \frac{1}{25} \). So, we write the equation as \( 5^{-2} = \frac{1}{25} \).

Key Concepts

Logarithms

Logarithms are an essential part of mathematics that help us express numbers that are quite large or quite small in a manageable way. They answer the question of "How many times must we multiply a number by itself to get another number?"
For example: If you ask "What is the logarithm base 5 of \( \frac{1}{25} \)?", you are thinking about how many times you multiply 5 together to get \( \frac{1}{25} \).

This can be written mathematically as \( \log_{5} \frac{1}{25} \). It equals -2 because multiplying 5 by itself twice in the negative (i.e., taking two reciprocal actions) gives \( \frac{1}{25} \).

• The base (5 in this case) determines how many times multiplication occurs.
• The argument (\( \frac{1}{25} \)) is the result of the multiplication.
• The logarithm (\(-2\)) is how many times you perform the multiplication based on the base.

Converting Logarithmic to Exponential Form

Converting a logarithmic equation into an exponential form is an important skill that simplifies interpretation and solution of equations. The general form of a logarithm equation is \( \log_{b} a = c \), where \( b \) is the base, \( a \) is the result after applying the base raised to \( c \), and \( c \) is the logarithmic value.
Let's look at the problem \( \log_{5} \frac{1}{25} = -2 \). It asks us to express it in exponential form. Using the definition of a logarithm, this equation means that 5 (the base) raised to the power of -2 gives \( \frac{1}{25} \) (the result).

To convert, we use the rule that \( b^{c} = a \):

• Identify base \( (b = 5) \).
• Find the logarithmic result \( (c = -2) \).
• Calculate \( 5^{-2} \) to get the argument \( (a = \frac{1}{25}) \).

Therefore, the exponential form is \( 5^{-2} = \frac{1}{25} \). Converting equations into exponential form often simplifies solving.

Properties of Exponents

Understanding the properties of exponents can greatly simplify computations. The problem asks us to understand what \( 5^{-2} \) means. Negative exponents like \(-2\) represent the reciprocal or inverse power of a number. This means instead of multiplying, you divide.

For any number \( x^{-n} \), where \( n \) is a positive integer:

• It equals \( \frac{1}{x^{n}} \). For instance, \( 5^{-2} \) is \( \frac{1}{5^{2}} \), which equals \( \frac{1}{25} \).
• Negative exponents "flip" the number from the numerator to the denominator, or vice versa.

These rules simplify computing negative exponents, making complex calculations approachable. When solving problems, checking these properties can provide a quick solution. Remember:
1. Positive exponents show multiplication.
2. Negative exponents reveal division or the reciprocal.