Problem 562 - Precalculus

Question

Solve for \(x\) by converting the logarithmic equation \(\log _{\frac{1}{7}}(x)=2\) to exponential form.

Step-by-Step Solution

Answer

The solution for \(x\) is \(\frac{1}{49}\).

Step 1: Identify the Logarithmic Format

The equation given is \(\log_{\frac{1}{7}}(x) = 2\). This is a logarithmic equation where \(\log_b(a) = c\) implies that \(b^c = a\).

Step 2: Convert to Exponential Form

Use the definition of logarithms to convert the equation. Since \(\log_{\frac{1}{7}}(x) = 2\), this converts to \(\left(\frac{1}{7}\right)^2 = x\).

Step 3: Calculate \(\left(\frac{1}{7}\right)^2\)

Calculate \(\left(\frac{1}{7}\right)^2\) which is \(\frac{1}{7} \times \frac{1}{7} = \frac{1}{49}\).

Step 4: State the Solution

Based on the conversion, \(x = \frac{1}{49}\).

Key Concepts

Exponential Form

The exponential form is a way of expressing a mathematical statement where a number, called the base, is raised to a power. This form is particularly helpful when dealing with logarithmic equations because it allows you to translate a logarithmic expression into a simpler mathematical structure. In our exercise, we started with a logarithmic equation \(\log_{\frac{1}{7}}(x) = 2\). Here, the base is \(\frac{1}{7}\), and the expression is saying "to what power should \(\frac{1}{7}\) be raised to yield \(x\)?" By converting it to exponential form, we write this as \((\frac{1}{7})^2 = x\). This conversion allows us to standardize and simplify our calculations.

Logarithm Conversion

Logarithm conversion is the process of translating a logarithmic equation into an exponential equation, making it easier to solve for unknown variables. This is based on the fundamental relationship between exponents and logarithms. If you have a logarithm \(\log_b(a)=c\), it can be converted to exponential form as \(b^c = a\). This relationship is crucial for solving equations that involve unknowns with logarithmic expressions.

• Identify the base \(b\), the result \(a\), and the exponent \(c\) in the given logarithmic equation.
• Transpose these elements into the exponential form \(b^c = a\).

For the problem \(\log_{\frac{1}{7}}(x) = 2\), the conversion process translates it into \((\frac{1}{7})^2 = x\), providing a straightforward way to find the value of \(x\).

Mathematical Calculation

Mathematical calculation involves computing numerical expressions using arithmetic operations. Once we have converted the logarithm into an exponential form, we need to perform calculations to find the solution. For our specific problem, we need to compute \((\frac{1}{7})^2\), which involves multiplying the fractional base by itself:

• Calculate: \((\frac{1}{7}) \times (\frac{1}{7}) = \frac{1}{49}\).
• This computation leads us to find that \(x = \frac{1}{49}\).

Calculations like this allow us to complete the conversion process from a logarithmic expression to a numerical result, making the problem-solving process concise and effective.