Problem 55 - 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. $$ \log _{2}(x+25)=4 $$

Step-by-Step Solution

Answer

The exact solution of the logarithmic equation is \(x = -9\).

Step 1: Rewrite in exponential form

To find the unknown \(x\), we rewrite the logarithmic equation in exponential form. From the equation, \(\log _{2}(x+25)=4\), by using the definition of the logarithmic function, we have \(2^4 = x + 25\).

Step 2: Solve for x

Next, we solve for the variable \(x\). So we perform the arithmetic operation \(2^4 - 25\) which equals \(16 - 25 = -9\). Hence, \(x = -9\).

Step 3: Check the solution within the domain

Lastly, we verify the solution within the original logarithmic function's domain. A logarithmic function is defined only for positive real numbers, which means \(x + 25\) must be greater than zero. However, when we substitute \(-9\) for \(x\), we get \(x + 25 = -9 + 25 = 16\), which is a positive real number. Therefore, the value of \(x\) obtained is valid.

Key Concepts

Exponential Form in Logarithmic Equations

Logarithmic equations often need to be transformed into exponential form to be solved. This is crucial because it simplifies the process of finding the unknown variable. To convert a logarithmic equation to its exponential form, we use the definition of logarithms: if \( \log_b(a) = c \), then this is equivalent to \( b^c = a \).
For example, in the equation \( \log_2(x+25) = 4 \), we convert it into exponential form as follows:

• The base of the logarithm is 2.
• The result we get when the base is raised to the power is 4.
• Thus, the exponential form is \( 2^4 = x + 25 \).

This rearrangement makes it straightforward to solve for \( x \). In this case, \( 2^4 = 16 \) is calculated, leading us directly to solve for the variable \( x \).
Converting a logarithmic equation into an exponential form is a powerful method for solving complex equations and a foundational concept in algebra.

Domain of Logarithmic Functions

The domain of logarithmic functions is essential to understand when solving logarithmic equations. A logarithmic function is defined only for positive real numbers. That means inside the logarithmic function, the expression must be greater than zero. Otherwise, the logarithm will not be defined.
In our example problem, \( \log_2(x+25) \), the expression inside the logarithm \( (x+25) \) must be positive:

• The inequality for the domain is \( x + 25 > 0 \).
• Solving this gives \( x > -25 \).

This tells us that \( x \) must be greater than -25 for the logarithmic expression to be valid.
It is crucial always to check this domain condition when solving logarithmic equations. Even if you find a solution for \( x \), it must be checked to ensure it fits within the domain constraints of the original equation. This prevents obtaining invalid solutions and understanding potential restrictions of the problem.

Solving Logarithmic Equations

Solving logarithmic equations involves several steps that ensure you find valid solutions that adhere to the domain conditions. Start by converting the logarithmic equation into exponential form, as mentioned earlier. This step will reduce the equation to a basic algebraic expression, which is typically easier to solve.
For example, in our specific problem:

• Converted to exponential form: \( 2^4 = x + 25 \)
• Calculate \( 16 = x + 25 \).

Subsequently, solve for \( x \) by isolating it:

• Subtract 25 from both sides to get \( x = 16 - 25 \).
• This results in \( x = -9 \), which must be checked against the logarithmic function's domain.

Finally, verify the found solution meets all domain requirements. If \( x + 25 = 16 \), which is a positive value, then \( x = -9 \) confirms it is within the valid range.
This systematic approach ensures that you not only solve the equation but also adhere to mathematical rules governing logarithmic expressions. Always remember that checking the domain is an integral part of solving these equations.