Problem 22 - Algebra A Combined Function

Question

Solve. $$ 6^{x}=36 $$

Step-by-Step Solution

Answer

The solution is \(x = 2\).

Step 1: Understand the Equation

We are given the equation \(6^x = 36\). Our goal is to find the value of \(x\) that makes this equation true.

Step 2: Rewrite 36 as a Power of 6

Notice that 36 can be expressed as a power of 6. Since 36 is equal to \(6^2\) (because \(6 \times 6 = 36\)), we can rewrite the equation as \(6^x = 6^2\).

Step 3: Set the Exponents Equal

Since the bases are equal, we can set the exponents equal to each other. Therefore, \(x = 2\).

Step 4: Verify the Solution

Substitute \(x = 2\) back into the original equation to verify: \(6^2 = 36\). The solution is correct.

Key Concepts

Equations

An equation is a mathematical statement asserting the equality of two expressions. It typically includes two sides separated by an equal sign, like in the equation \(6^x = 36\). Here, the goal is to find the value of \(x\) that makes the equation true. Equations are versatile and can involve numbers, variables, and operations such as addition, subtraction, multiplication, division, or exponentiation.

When dealing with equations, we often solve them to find unknown values. This process requires manipulating the original equation to isolate the variable we need to solve for. In our exercise, the variable \(x\) was the unknown, and we achieved this by employing various mathematical strategies to simplify the equation.

Exponents

An exponent represents repeated multiplication of a base number. It is written as a small number to the upper right of the base, like in \(6^2\), which means \(6 imes 6\). Exponents are powerful tools in mathematics because they allow for expressing large numbers succinctly.

In the exercise \(6^x = 36\), the exponent is \(x\), and the base is \(6\). To solve such equations, understanding how to rewrite numbers in terms of exponents is crucial. By recognizing that 36 can also be expressed as a power of 6 (\(6^2\)), identical bases allow us to equate the exponents directly. This enables problem solvers to deduce the value of the unknown exponent in the equation.

Solving Equations

Solving equations involves finding the value of the unknown that satisfies the equation. In the provided exercise, we started by examining the equation \(6^x = 36\) and then transformed 36 into \(6^2\) to have a common base. Once both sides had the same base, comparing the exponents was straightforward.

The key strategy was to rewrite numbers to reveal the same base when dealing with exponential equations. This makes it easy to focus on the exponents, discarding the now-identical bases, and solve the simpler equation \(x = 2\).

When we substitute \(x = 2\) back into the original equation to verify, we confirm that the solution is correct. Thus, solving exponential equations often involves recognizing patterns and using properties of exponents effectively.