Problem 22 Solve each equation for $x .$ ... [FREE SOLUTION]

Chapter 12: Problem 22

Solve each equation for $x .$ See Example 4. $$ 6^{x}=36 $$

Short Answer

Expert verified

x = 2.

Step by step solution

Express the Right Side in terms of a Base

The right side of the equation is 36. Notice that 36 can be expressed as a power of the base 6: 36 = 6^2. Therefore, the equation can be rewritten as 6^x = 6^2.

Equate the Exponents

Since the bases on both sides of the equation are equal (both are 6), we can set the exponents equal to each other according to the property of equality of exponents: x = 2.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Solving Equations

Solving equations is a fundamental skill in algebra that involves finding the value of the variable that makes the equation true. When dealing with exponential equations, the goal is to isolate the variable, often located in the exponent. Begin by ensuring both sides of the equation are expressed in terms of the same base. This simplifies the problem, allowing you to focus on the exponent where the variable resides. For example, in the equation $6^x = 36$, it's beneficial to rewrite 36 as $6^2$ to utilize a common base on both sides. This converts the problem from a complex calculation into a simpler task involving exponent equality.

Equality of Exponents

The equality of exponents is a principle used to solve exponential equations when both sides of the equation have the same base. Once both sides are written with the same base, the equation implies that the exponents themselves must be equal for the equality to hold true.For example, once we rewrite $36$ as $6^2$ in the equation $6^x = 6^2$, the base 6 on both sides allows us to directly equate the exponents, leading to $x = 2$. This principle simplifies solving such equations and reduces the complexity involved. Having a common base is key to using this technique effectively.

Expressing Numbers as Powers of a Base

Expressing numbers as powers of a base is a crucial step in solving exponential equations. It involves rewriting or decomposing a number into a power that shares the same base as the exponential term you're dealing with.This technique is particularly handy because it simplifies equations and facilitates the application of the equality of exponents. For example, in the equation $6^x = 36$, notice that 36 can be decomposed into $6^2$. When you express numbers this way, solving for the variable becomes straightforward, as it allows for a direct comparison of exponents.

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91影视

Solve each equation for \(x .\) See Example 4. $$ 6^{x}=36 $$

Short Answer

Step by step solution

Express the Right Side in terms of a Base

Equate the Exponents

Key Concepts

Solving Equations

Equality of Exponents

Expressing Numbers as Powers of a Base

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