Problem 43 - Algebra and Trigonometry Real Mathematics, Real People

Question

Solve the logarithmic equation. $$\log _{11} x=-1$$

Step-by-Step Solution

Answer

The solution to the equation is \( x = \frac{1}{11} \).

Step 1: Convert the Logarithmic Equation to Exponential Form

The logarithmic equation \( \log _{11} x = -1 \) can be transformed into its equivalent exponential form. This is achieved through the general rule: If \( \log _{b} a = c \), then \( b^c = a \). By using this rule, the given equation becomes \( 11^{-1} = x \).

Step 2: Calculate the Value of x

To solve for x, calculate \( 11^{-1} \). This means raising 11 to the power of -1 or, in other words, calculate the reciprocal of 11. Thus, \( x = \frac{1}{11} \).

Key Concepts

Exponential Form

Understanding the exponential form is crucial when working with logarithmic equations. To convert a logarithmic equation into exponential form, you use a specific rule: If you have \( \log_b a = c \), it translates to \( b^c = a \). This equivalence helps in simplifying the expression and finding the value you need.In the given exercise \( \log_{11} x = -1 \), applying this rule means that we change the equation into the exponential form as \( 11^{-1} = x \). This step is foundational because it allows us to work directly with the base 11 raised to a power, making it easier to calculate and solve the equation.

Logarithmic Functions

Logarithmic functions are a type of mathematical relationship that relates an exponent to its base number. These functions are the inverse of exponential functions, meaning they are directly connected to the exponential expressions we often encounter.When working with logarithms, the function is generally defined as \( \log_b x \), where \( b \) is the base and \( x \) is the argument of the logarithm. The base \( b \) is the number we are multiplying, while the result is the power to which the base is raised to get \( x \).Understanding this inverse relationship is essential, especially since many logarithmic problems involve converting between logarithmic and exponential forms. With our example \( \log_{11} x = -1 \), the logarithm simplifies the process of determining \( x \) by translating it into \( 11^{-1} = x \), which then points directly to the solution.

Solving Equations

Solving equations, particularly logarithmic ones, requires a systematic approach. First, we transform the logarithmic equation into an exponential form. This switches our perspective and simplifies calculations.

• Start by identifying the base of the logarithm; in our problem it is 11.
• Next, use the power indicated on the right side of the equation; here, it is -1.
• Combine these to express the equation in exponential terms, \( 11^{-1} \).

Now, solving \( 11^{-1} \) means finding \( x \) in terms of familiar numerical operations. Calculate the reciprocal of 11, as raising a number \( x \) to the power of -1 is the same as finding its reciprocal. With this knowledge, the solution is given by \( x = \frac{1}{11} \).Approaching logarithmic equations by converting them first simplifies the steps and reduces potential errors, allowing for straightforward calculations.