Problem 15 - Algebra A Combined Function

Question

Write each as an exponential equation. $$ \log _{3} \frac{1}{81}=-4 $$

Step-by-Step Solution

Answer

The exponential equation is \( 3^{-4} = \frac{1}{81} \).

Step 1: Understand the Logarithmic Equation

We start with the logarithmic equation: \( \log_{3} \frac{1}{81} = -4 \). This equation states that the log base 3 of \(\frac{1}{81}\) equals \(-4\).

Step 2: Recall the Definition of a Logarithm

The definition of a logarithm \( \log_{b}(a) = c \) is equivalent to the exponential form \( b^c = a \). Here, \( b \) is the base, \( a \) is the result, and \( c \) is the exponent.

Step 3: Write the Exponential Equation

Using the definition of a logarithm, convert \( \log_{3} \frac{1}{81} = -4 \) into an exponential equation: \( 3^{-4} = \frac{1}{81} \).

Step 4: Verify Your Conversion

Ensure the exponential equation \( 3^{-4} = \frac{1}{81} \) is correct. Evaluate \( 3^{-4} \): it's equal to \( \frac{1}{3^4} \), which simplifies to \( \frac{1}{81} \), thus matching the original equation.

Key Concepts

Logarithmic Equations

Logarithmic equations are equations that involve the logarithm of a variable or expression. They are typically written in the form \(\log_{b}(a) = c\), where \(b\) is the base of the logarithm, \(a\) is the argument or result, and \(c\) is the exponent. Understanding these equations requires knowing what a logarithm does. Essentially, a logarithm answers the question: "To what power must the base \(b\) be raised, to obtain \(a\)?"

In the exercise, \(\log_{3} \frac{1}{81} = -4\), the equation states that \(3\) raised to what power equals \(\frac{1}{81}\)? The answer is \(-4\).

Key points to remember about logarithmic equations are:

• The base \(b\) must be a positive number, and it cannot be \(1\).
• The argument \(a\) must be positive as well, because logarithms of non-positive numbers are undefined in the real number system.
• Logarithmic functions are the inverse of exponential functions.

This inverse relationship is crucial to solving logarithmic equations by converting them to exponential equations, which often makes the problem easier to solve.

Conversion Between Logarithmic and Exponential Forms

Logarithmic and exponential forms are two sides of the same mathematical coin. Being able to convert between these forms is very useful as it allows you to solve problems more easily.

The logarithmic form \(\log_{b}(a) = c\) can be rewritten as an exponential form \(b^c = a\). Let's break it down:

• \(b\) is the base. It remains the same in both the logarithmic and exponential forms.
• \(c\) is the power or exponent in the exponential form.
• \(a\) is the result you want to achieve with your base and exponent.

Our exercise illustrates this conversion beautifully. We have \(\log_{3} \frac{1}{81} = -4\), meaning in exponential form, \(3^{-4} = \frac{1}{81}\), demonstrating that \(3\) raised to the power of \(-4\) equals \(\frac{1}{81}\).

By practicing this conversion, you can easily shift between forms to tackle different types of problems efficiently. It can also aid in understanding how growth processes work, such as in compound interest or population growth, where exponential functions often play a key role.

Properties of Exponents

Exponents are a way to represent repeated multiplication of a number by itself. The rules governing exponents are simple but powerful and apply in various mathematical contexts, including exponential equations related to logarithms.

Here are some important properties of exponents that can help in solving such problems:

• Negative exponent: \(b^{-c} = \frac{1}{b^c}\). A negative exponent essentially "flips" the base into its reciprocal. In the exercise, \(3^{-4} = \frac{1}{81}\) is a perfect example of this property.
• Zero exponent: Any base raised to the zero power equals one: \(b^0 = 1\).
• Multiplication of powers: When multiplying with the same base, add the exponents: \(b^m \cdot b^n = b^{m+n}\).
• Power of a power: When raising a power to another power, multiply the exponents: \((b^m)^n = b^{mn}\).
• Division of powers: When dividing with the same base, subtract the exponents: \(\frac{b^m}{b^n} = b^{m-n}\).

By understanding these properties, you can quickly manipulate exponential expressions, making it easier to both check your work and solve related equations effectively. Mastery of these properties can significantly enhance your mathematical problem-solving skills.