Problem 36 - Algebra 2 and Trigonometry

Question

Solve each equation and check. \(5+7^{x}=6\)

Step-by-Step Solution

Answer

The solution is \(x = 0\).

Step 1: Isolate the Exponential Term

First, subtract 5 from both sides of the equation to isolate the exponential term. This gives us:\[7^{x} = 6 - 5\]Simplifying the right side, we find:\[7^{x} = 1\]

Step 2: Solve the Exponential Equation

Since the equation is now \(7^{x} = 1\), we need to find the value of \(x\) that makes this true. Recall that any non-zero number raised to the power of 0 is 1. Thus:\[7^{x} = 7^{0}\]This implies that:\[x = 0\]

Step 3: Check the Solution

Substitute \(x = 0\) back into the original equation to verify:\[5 + 7^{0} = 6\]Since \(7^{0} = 1\), the left-hand side becomes:\[5 + 1 = 6\]This simplifies to \(6 = 6\), confirming our solution is correct.

Key Concepts

Isolating the Exponential Term

When faced with an exponential equation, the initial task is to isolate the exponential term, which is vital in simplifying the problem. For instance, in the equation we began with:

• We had: \(5 + 7^x = 6\)
• To isolate the exponential term \(7^x\), subtract 5 from both sides, leaving: \(7^x = 6 - 5\).
• The equation simplifies to \(7^x = 1\).

Isolating the exponential term transforms the equation into a form that is ready to be solved, setting the stage for the next steps. It also simplifies the algebraic landscape, making it much easier to analyze and solve.

Exponential Equation

Once the exponential part of your equation is isolated, you encounter what is known as an exponential equation. In our context, we have the simplified equation:

• \(7^x = 1\).

To solve this, recall the rule: any non-zero number raised to the power of zero is 1. This incredible property of exponents significantly aids in solving exponential equations:

• Since \(7^0 = 1\), it implies \(x = 0\) right away.

Understanding this fundamental property of exponents is pivotal, as it allows us to find the solution almost immediately without further calculations. Exponential equations often hide solutions that are rooted in these simple exponent rules, making familiarity with these rules invaluable.

Checking Solutions

Even after solving an exponential equation, it's crucial to check your solution to ensure its accuracy. Substituting your result back into the original equation is a foolproof way to verify it. Here’s how we did it:

• Substitute \(x = 0\) back into the original equation: \(5 + 7^0 = 6\).
• Recognize that \(7^0 = 1\), so the equation becomes: \(5 + 1 = 6\).
• This simplifies to \(6 = 6\), confirming that our solution is indeed correct.

This checking step serves as a confirmation that the solution not only satisfies the equation, but also reinforces the understanding of exponential properties. Always remember that the verification step is an essential component of solving equations, especially for catching potential errors early.