Problem 322 - Precalculus

Question

Use the definition of a logarithm to solve the equation. \(-8 \log _{9} x=16\)

Step-by-Step Solution

Answer

The solution is \( x = \frac{1}{81} \).

Step 1: Isolate the Logarithm Term

Start by dividing both sides of the equation by -8 to isolate the logarithm term. This results in:\[ \log_{9} x = \frac{16}{-8} \]Simplifying the right side gives:\[ \log_{9} x = -2 \]

Step 2: Convert Logarithmic Form to Exponential Form

Use the definition of a logarithm, which states that if \( \log_b a = c \), then \( b^c = a \). Applying this definition, convert the equation \( \log_{9} x = -2 \) to its exponential form:\[ 9^{-2} = x \]

Step 3: Simplify the Exponential Expression

Calculate what \( 9^{-2} \) is by finding the reciprocal of \( 9^2 \):\[ 9^2 = 81 \]So, \( 9^{-2} = \frac{1}{81} \). Therefore, \( x = \frac{1}{81} \).

Key Concepts

Understanding Exponential Form

The concept of exponential form is an essential part of understanding logarithms. A logarithm is essentially the inverse operation to exponentiation. When you have an expression like \( \log_b a = c \), it means you are looking for the power \( c \) that you must raise the base \( b \) to in order to get \( a \). In other words, it gives us the equation \( b^c = a \).

This conversion between forms is crucial when solving logarithmic equations. For example, in the exercise, we had \( \log_{9} x = -2 \). By applying the exponential form, we rewrite the equation as \( 9^{-2} = x \).

Converting from logarithmic to exponential form simplifies the process of finding the unknown variable, which in this case is \( x \). It turns a complicated logarithmic expression into a simpler exponential one, making it easier to solve.

Exploring the Concept of Reciprocals

A reciprocal is a concept often used in various areas of mathematics, including when dealing with exponents. In simple terms, the reciprocal of a number is what you multiply the original number by to get 1. If you have a number \( a \), its reciprocal is \( \frac{1}{a} \).

In the context of the exercise, when we dealt with \( 9^{-2} \), we were finding the reciprocal of \( 9^2 \). This is because a negative exponent represents the reciprocal of the base raised to the positive exponent. So, for \( 9^{-2} \), we find \( 9^2 = 81 \) and then take its reciprocal, which is \( \frac{1}{81} \).

Understanding reciprocals helps in simplifying expressions and solving equations, especially when negative exponents are involved.

Simplifying Rational Expressions

Simplifying rational expressions involves reducing a fraction or an algebraic expression to its simplest form. This is done by canceling common factors in the numerator and the denominator, or by simplifying terms as much as possible.

In the given exercise's solution, when we find that \( 9^2 = 81 \), we express \( 9^{-2} \) as \( \frac{1}{81} \). This step is a straightforward example of turning a more complex power expression with a negative exponent into a simple fraction, making it easier to understand and use.

It's important to simplify expressions whenever possible because it helps in making mathematical problems more manageable, both in solving equations and when working with more complex expressions as part of larger problems. The simpler the expression, the easier it is to see what's happening mathematically and to solve or further manipulate it.