Problem 32 - College Algebra

Question

Solve each logarithmic equation in Exercises \(27-44 .\) Be sure to reject any value of \(x\) that produces the logarithm of a negative number or the logarithm of \(0 .\) $$\log _{7}(x+2)=-2$$

Step-by-Step Solution

Answer

The solution to the logarithmic equation \( \log_7{(x+2)} = -2 \) is \( x = -95/49 \)

Step 1: Apply exponential form

To get rid of the logarithm, convert the equation from logarithmic form to exponential form using the property \( \log_b{a}=c \to b^c = a \). This gives: \( 7^{-2} = x + 2 \), which simplifies to \( 1/49 = x + 2 \).

Step 2: Solve for x

Isolate \(x\) by subtracting 2 from both sides of the equation: \( x = 1/49 - 2 \) which simplifies to \( x = -95/49 \)

Step 3: Check the solution

Substitute \( x = -95/49 \) into the original equation and check if the equation is satisfied, also ensure that the value of \( x \) does not result in a logarithm of a negative number or zero. Substituting \( x = -95/49 \) back into \( \log_7{x+2} = -2 \) gives \( \log_7{-95/49 +2} = -2 \). This simplifies to \( \log_7{1/49} = -2 \), and since this is indeed true, \( x = -95/49 \) is a valid solution to the equation.

Key Concepts

Logarithmic to Exponential Form

Understanding how to transform a logarithmic equation into its exponential form is crucial in solving logarithmic equations. A logarithmic equation is of the form \( \log_b(a) = c \). To convert it to exponential form, we use the basic definition of logarithms: if \( b \) is a positive real number not equal to 1, \( a > 0 \), and \( c \) is a real number, then \( b^c = a \) is the equivalent exponential equation. In the example \( \log_7(x+2)=-2 \), it is transformed into \( 7^{-2} = x + 2 \) by applying this principle. This switch from log to exponential form provides a more straightforward pathway to isolate the variable and solve for it.

When working with logarithms, it's also crucial to remember that the argument of the logarithm—that is, the \( x \) in \( \log_b(x) \)—must always be positive since the logarithm of a negative number or zero is undefined. So, in our exercise, we have to ensure that \( x+2 \) remains positive before considering any solution as valid.

Logarithm Properties

There are several properties of logarithms that can simplify the process of solving logarithmic equations:

• Product Rule: \( \log_b(mn) = \log_b(m) + \log_b(n) \), which means that the logarithm of a product is the sum of the logarithms.
• Quotient Rule: \( \log_b(\frac{m}{n}) = \log_b(m) - \log_b(n) \), which tells us that the logarithm of a quotient is the difference of the logarithms.
• Power Rule: \( \log_b(m^n) = n\log_b(m) \), indicating that the logarithm of a power is the exponent times the logarithm of the base.
• Change-of-Base Formula: allows you to rewrite a logarithm in terms of logs with a different base which is particularly helpful when using calculators.

Understanding these properties can help with more complicated logarithmic equations where combining or converting logs is necessary to isolate the variable.

Logarithmic Equations Steps

Solving logarithmic equations typically involves a sequence of steps that will lead to the solutions. Here’s a basic approach illustrated by the given example:

Step 1: Begin by converting the logarithmic equation into its exponential form, which will often make the equation easier to handle. For the sample problem \( \log_7(x+2)=-2 \), we convert it to \( 7^{-2} = x + 2 \).

Step 2: Solve for the variable. In the example, we isolate \( x \) by subtracting 2 from both sides to get \( x = 1/49 - 2 \).

Step 3: Always verify the solution by plugging it back into the original equation. The solution needs to satisfy the equation and the argument of the logarithm should not be negative or zero. With \( x = -95/49 \), we check the solution against the original logarithmic form to confirm its validity.