Problem 57 - College Algebra

Question

Solve each logarithmic equation in Exercises \(49-92 .\) Be sure to reject any value of \(x\) that is not in the domain of the original logarithmic expressions. Give the exact answer. Then, where necessary, use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. $$ \log _{3}(x+4)=-3 $$

Step-by-Step Solution

Answer

The solution (\(x = 3^{-3} - 4\)) doesn't satisfy the domain restrictions of the original logarithmic equation. If a decimal approximation were calculated, it still wouldn't respect the domain.

Step 1: Understand the logarithm equation

The given equation is \(\log _{3}(x+4)=-3 \). This is a logarithmic equation with base 3. The idea is to eliminate the logarithm in order to make \(x\) easier to isolate.

Step 2: Apply the definition of the logarithm

The definition of a logarithm can be written as: if \( \log_b a = c \), then \( b^c = a \). Thus, we can write the equation in exponential form: \(3^{-3} = x + 4\).

Step 3: Solve for \(x\)

Solving equation for \(x\), we obtain: \(x = 3^{-3} - 4 \).

Step 4: Check the solution and domain

In logarithmic equations, it's important to ensure that the solution respects the domain of the original logarithmic function. Logarithms are undefined for negative numbers. So, \(3^{-3} - 4\) should be greater than 0. However, it is not. Therefore, this solution is not valid in the original logarithmic expression.

Step 5: Decimal approximation

Though our value for \(x\) is invalid in the original domain, if required, we can still find the decimal approximation (correct up to two decimal places). Thus, the decimal expression of \(x=3^{-3} - 4 \) can be calculated using a calculator.

Key Concepts

Logarithmic Functions

Logarithmic functions are the inverses of exponential functions, and they play a crucial role in solving equations where the variable is located in an exponent. A logarithmic function is typically written as \( \text{log}_b (x) \), where \( b \) is the base and \( x \) is the argument of the logarithm. To understand how to work with them, it's important to remember the definition: if \( \text{log}_b (a) = c \), then the equivalent exponential form is \( b^c = a \).

When solving an equation such as \( \text{log}_3 (x+4) = -3 \), one applies this definition to rewrite the logarithm in its exponential form to isolate the variable. Throughout the process, maintaining an awareness of the properties and behaviors of logarithmic functions is key to arriving at the correct solution. As they are inverse functions, logarithms have a unique ability to 'bring down' exponents, making them indispensable in various fields such as mathematics, science, and engineering.

In educational contexts, a solid grasp of logarithmic functions is essential not merely for solving textbook exercises but for understanding the logarithmic scale, which is used in measuring phenomena that have a wide range, such as earthquake intensities and acidity in pH.

Exponential Form

Exponential form is a way of expressing numbers as a base raised to a power. It simplifies dealing with numbers that grow or decay rapidly. In the context of logarithmic equations, the exponential form becomes a strategic step in the process of solving for the variable. For instance, in the exercise \( \text{log}_3 (x+4) = -3 \), to remove the logarithm and expose the variable, one converts it to exponential form which results in \( 3^{-3} = x + 4 \).

It's essential for students to be comfortable switching between logarithmic and exponential forms because it's a fundamental part of solving logarithmic equations. The ability to do this also reinforces the understanding of the inverse relationship between logarithms and exponents. By converting to the exponential form, we take a step away from the complexity of logarithms and use more familiar algebraic operations to solve for the variable.

Logarithm Domain Restrictions

One of the crucial aspects of logarithms that must not be overlooked is domain restrictions. The domain of a logarithmic function is the set of all possible values of \( x \) that make the function valid and real. The rule of thumb is that the argument of a logarithm must always be greater than zero. This is because you cannot take the logarithm of a non-positive number.

In our exercise \( \text{log}_3 (x+4) = -3 \), after finding that the solution to the equation is \( x = 3^{-3} - 4 \), it is necessary to verify whether this value adheres to the domain restrictions of the original logarithmic expression. Since the result of this calculation is a negative number and a logarithm cannot have a negative argument (\( x+4 \)), the solution is rejected. The concept of domain is vital, not just in logarithmic equations but in all branches of mathematics, as it determines the validity of solutions. Educators must prioritize teaching about domain restrictions to ensure students develop a comprehensive understanding of not just how to solve equations, but why certain solutions may or may not be viable, based on the properties of the functions involved.