Problem 137 - College Algebra Essentials

Question

Determine whether each statement makes sense or does not make sense, and explain your reasoning. Because the equations \(2^{x}=15\) and \(2^{x}=16\) are similar, I solved them using the same method.

Step-by-Step Solution

Answer

Yes, the statement makes sense. Although the specific solutions are different, the same method, which involves using the properties of logarithms, is applicable to solve both given equations \(2^{x}=15\) and \(2^{x}=16\).

Step 1: Understanding the problem

The statement in question is examining if methods of solutions can be applied uniformly across similar problems. The two equations given, \(2^{x}=15\) and \(2^{x}=16\), indeed have the same structure, both being exponential equations with base 2. The similarity ends there though as the right hand side of the equations are different. This influences the exact solution but does not impact the methodology to solve them.

Step 2: Explaining the solution method

Exponential equations like the ones given can be solved using the property of logarithms. The general process involves taking the logarithm of both sides of the equation, often base 2 logarithm when the base of the exponential is 2. This will transform the equation into a format that isolates \(x\) and makes the solution straightforward. For the equation \(2^{x}=15\), the solution will be \(x = \log_{2}15\). Similarly, for the equation \(2^{x}=16\), the solution will be \(x = \log_{2}16 = 4\).

Step 3: Concluding the reasoning

The method to solve both equations is indeed the same, rooted in taking advantage of the properties of logarithms. The specific solution steps differ slightly between the two equations due to the different numbers on the right hand side of the equations. So the statement does make sense, because the equations are similar enough to use the same method to solve them.