Problem 12 - Algebra 2

Question

Write each equation in logarithmic form. $$ \left(\frac{1}{3}\right)^{3}=\frac{1}{27} $$

Step-by-Step Solution

Answer

\( \log_{1/3} \frac{1}{27} = 3 \)

Step 1: Identify the base, exponent, and result

In the given exponential equation, \( \left(\frac{1}{3}\right)^{3}=\frac{1}{27} \), the base is \( \frac{1}{3} \), the exponent is 3 and the result or the number that the equation equates to is \( \frac{1}{27} \).

Step 2: Convert to Logarithmic form

The logarithmic form for a given exponential equation b^y = x is given by \( \log_{b} x = y \). Here, b is the base, x is the result or the number that the equation equates to, and y is the exponent. Substituting the given values from Step 1 we get \( \log_{1/3} \frac{1}{27} = 3 \). This is the logarithmic form for the given equation.

Key Concepts

Exponents

Exponents are mathematical expressions that tell us how many times to multiply a number by itself. In the expression \( a^n \), \( a \) is the base and \( n \) is the exponent. For example, \( 2^3 \) means \( 2 \times 2 \times 2 \), which equals 8.
Understanding exponents makes it easier to deal with large numbers since repeated multiplication can be simplified to multiplication with a single operation. When working with exponents, it’s important to remember these rules:

• Any number raised to the power of 1 is the number itself (e.g., \( a^1 = a \)).
• Any number raised to the power of 0 is 1, provided that the base is not zero (e.g., \( a^0 = 1 \)).
• A negative exponent indicates reciprocal (e.g., \( a^{-n} = \frac{1}{a^n} \)).

When you have a fraction like \( \left(\frac{1}{3}\right)^3 = \frac{1}{27} \), the exponent tells you to multiply \( \frac{1}{3} \) three times. In simpler terms, \( \frac{1}{3} \times \frac{1}{3} \times \frac{1}{3} = \frac{1}{27} \).
These rules and this understanding are crucial when converting exponential expressions to logarithmic forms.

Logarithms

Logarithms are the inverse operations of exponents. Essentially, they answer the question: "To what power must the base be raised, to produce a certain number?"
A logarithm has the form \( \log_{b} x = y \), which reads as "the log base \( b \) of \( x \) equals \( y \)." This means that \( b^y = x \).
Let's look at an example: if \( 3^2 = 9 \), then \( \log_{3} 9 = 2 \), because 3 raised to the power of 2 is 9. Logarithms have several properties that make them useful:

• \( \log_{b} 1 = 0 \) for any positive \( b \), because any number raised to the power 0 is 1.
• \( \log_{b} b = 1 \) because any number raised to the power 1 is itself.
• \( \log_{b} (xy) = \log_{b} x + \log_{b} y \) (useful for multiplying numbers).
• \( \log_{b} \left(\frac{x}{y}\right) = \log_{b} x - \log_{b} y \) (useful for division).
• \( \log_{b} (x^r) = r \cdot \log_{b} x \) (useful for exponentiation).

Understanding these properties helps you easily convert between exponential and logarithmic forms without confusion. It’s a powerful tool in both simple and complex mathematical contexts.

Base of a Logarithm

The base of a logarithm is a crucial component that determines the scale of measurement for the logarithm. In the expression \( \log_{b} x = y \), \( b \) is the base. The base tells us what number is "going up" in powers to end up at the value \( x \).
For instance, in \( \log_{2} 8 = 3 \), the base is 2. It means you must raise 2 to the power of 3 to get 8, because \( 2^3 = 8 \).
When dealing with logs, different bases might be used:

• Natural logarithms use the base \( e \), an irrational number approximately equal to 2.718. They are common in calculus and continuous growth models.
• Base 10 logarithms, also known as common logarithms, are often used in scientific contexts because they are easy to work with (e.g., \( \log_{10} \)).
• Logarithms with a base of 2 are prevalent in computer science, notably because binary systems operate on base 2.

In our specific problem, the base is \( \frac{1}{3} \). This means in logarithmic terms, we're finding what power of \( \frac{1}{3} \) makes \( \frac{1}{3}^3 \) equal to \( \frac{1}{27} \). By recognizing the base, we unlock the relationship between the exponential and logarithmic forms of the expression.