Problem 46 - Algebra 2

Question

Solve each equation. Check your solutions. \(\log _{\frac{1}{7}} x=-1\)

Step-by-Step Solution

Answer

The solution is \(x = 7\).

Step 1: Understand the Logarithmic Equation

We need to solve the equation \(\log_{\frac{1}{7}} x = -1\). A logarithm of a base \(b\) for a number \(x\) equals \(y\) means that \(b^y = x\). Here, the base is \(\frac{1}{7}\), and the result is \(-1\). This means \(\left(\frac{1}{7}\right)^{-1} = x\).

Step 2: Convert to an Exponential Equation

Convert the logarithmic equation to its equivalent exponential form. The equation \(\log_{\frac{1}{7}} x = -1\) can be rewritten as \(x = \left(\frac{1}{7}\right)^{-1}\).

Step 3: Evaluate the Exponential Expression

Calculate \(\left(\frac{1}{7}\right)^{-1}\). An exponent of \(-1\) means taking the reciprocal of the base: \(\left(\frac{1}{7}\right)^{-1} = 7\). Therefore, \(x = 7\).

Step 4: Verify the Solution

Verify by substituting \(x = 7\) back into the original logarithmic equation: \(\log_{\frac{1}{7}} 7 = -1\). Since \(\left(\frac{1}{7}\right)^{-1} = 7\) holds true, the solution is correct.

Key Concepts

Exponential Equations Simplified

Exponential equations are equations where variables appear as exponents. These equations typically take the form of \(a^x = b\), and solving them involves finding the value of the variable. Here, the base \(a\) is raised to a power represented by the unknown variable \(x\).
A useful approach to solve these equations is using logarithms. Logarithms allow you to express an exponent in terms of multiplication or division. When you convert a logarithmic equation to exponential form, you are essentially expressing the problem as a more familiar arithmetic equation.
For example, in the problem, we initially had \(\log_{\frac{1}{7}} x = -1\). By converting it to its exponential form, we obtained \(x = \left(\frac{1}{7}\right)^{-1}\). This makes it easier to solve. Understanding this conversion is crucial for tackling exponential equations effectively.

The Art of Reciprocal Calculation

Reciprocal calculation is a fundamental skill when working with negative exponents or numbers in fraction form. The reciprocal of a number is one divided by that number. For instance, the reciprocal of \( \frac{1}{7} \) is \( 7 \).
Negative exponents indicate that you should take the reciprocal of the base and then raise it to the absolute value of the exponent. In our example, \(\left(\frac{1}{7}\right)^{-1}\) is computed by swapping the numerator and the denominator, resulting in \(7\).
Calculating reciprocals is a straightforward but essential operation for solving problems involving negative exponents. It turns a division problem into a multiplication one, which often simplifies the process. Remember, by using reciprocal calculation, you're making complex equations much more manageable.

Understanding the Base of a Logarithm

The base of a logarithm is a critical component in any logarithmic equation. It is the number that is raised to a power in the equation: \(b^y = x\). The choice of base determines how the equation is solved and interpreted.
In this exercise, the base of the logarithm is \(\frac{1}{7}\). This means that we're looking for a number \(x\) such that \(\left(\frac{1}{7}\right)^y = x\). Understanding and identifying the base is vital, as it allows you to rewrite logarithmic equations in an exponential format for easy computation.

• A base greater than 1 typically produces positive logarithmic values.
• A base between 0 and 1, like \(\frac{1}{7}\), will create negative logarithmic values when solving equations.

Correctly identifying the base helps you understand why the logarithmic value is positive or negative and guides you toward finding a suitable solution. This is the backbone of solving logarithmic equations efficiently.