Problem 76 - Introductory and Intermediate Algebra for College Students 4th

Question

Solve each logarithmic equation. 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. $$\log _{2}(x-5)=\log _{2} 4$$

Step-by-Step Solution

Answer

The solution to the logarithmic equation \(\log _{2}(x-5)=\log _{2} 4\) is \(x = 9\).

Step 1: Identify the Base

Both sides of the equation have the same base, which is 2. This condition allows for a straightforward solution.

Step 2: Use the Logarithmic Equality Rule

Because both sides of the equation use the same base, one can set the arguments of the logarithm equal to each other. Thus, we get \(x-5 = 4\).

Step 3: Solve for x

Solving this equation for \(x\), we get \(x = 4 + 5 = 9\).

Step 4: Check the Domain

Make sure that the solution \(x = 9\) lies within the domain of the original logarithmic function. For the function \(\log _{2}(x-5)\), \(x-5\) must be greater than zero, which means \(x\) must be greater than 5. Since our solution \(x = 9\) is indeed greater than 5, it is a valid solution, and it is not rejected.

Key Concepts

Base of Logarithm

In logarithmic equations, the base of the logarithm is crucial as it determines the relationship between the exponent and its corresponding power. A logarithm like \(\log_2(x-5)\) means we are investigating the power to which 2 (the base) must be raised to obtain \(x-5\). The base, in this case, is 2, which matches on both sides of the equation in our exercise.
The base dictates the potential transformations and the simplicity of solving such equations. If the bases of the logarithms on both sides are the same, you can directly equate their arguments (the expression inside the logarithm), significantly simplifying the solving process.

Domain of Logarithmic Function

The domain of a logarithmic function is the set of all possible input values (x-values) that meet the condition making the logarithm a valid operation, which generally means positive values.
For the exercise \(\log_2 (x-5) = \log_2 4\), the argument \(x-5\) must be greater than 0 for the logarithm to be defined. This requires \(x > 5\). For any solution to this logarithmic equation, we need to verify that the output value satisfies this domain condition. If not, the answer must be rejected because it would lead to a mathematically undefined situation.

Logarithmic Equality Rule

The foundational principle for solving logarithmic equations is the Logarithmic Equality Rule. When two logarithms with the same base are equal, their arguments must also be equal. This rule simplifies the equation by dropping the logarithm, allowing us to solve the resulting simple algebraic equation.
In our example, \( \log_2(x-5) = \log_2 4 \,\) both sides have the base 2. Thus, we set the inside expressions equal to each other: \( x-5 = 4 \.\) This step directly leads us to a solvable equation, making this mathematical property hugely beneficial in finding solutions to logarithmic problems.

Decimal Approximation

Often, solving logarithmic equations yields exact solutions. However, when required, you might need to provide a decimal approximation, particularly for complex expressions or when the problem demands it.
For equations like the one presented, \( \log_2(x-5) = \log_2 4 \,\) our exact solution was \( x = 9 \,\) which doesn't need further rounding. However, in cases where this is necessary, use a calculator. Make sure to round your final result properly, typically to two decimal places, for clarity and precision.