Problem 27 - Algebra 2 and Trigonometry

Question

Solve each equation and check. \(100^{x}=1,000^{x-1}\)

Step-by-Step Solution

Answer

The solution is \(x = 3\).

Step 1: Recognize the Relationship Between Bases

Notice that 100 and 1,000 can be expressed as powers of 10. We have 100 as \(10^2\) and 1,000 as \(10^3\). Use this knowledge to rewrite the equation: \( (10^2)^x = (10^3)^{x-1} \).

Step 2: Simplify Using Power of a Power Rule

Apply the power of a power rule which states \((a^m)^n = a^{m \times n}\). Simplify the equation: \(10^{2x} = 10^{3(x-1)}\).

Step 3: Equate the Exponents

Since the bases are identical, equate their exponents: \(2x = 3(x-1)\).

Step 4: Solve the Linear Equation

Distribute the 3 on the right-hand side: \(2x = 3x - 3\). Bring the variable terms to one side by subtracting \(3x\) from both sides: \(-x = -3\). Divide by -1 to solve for \(x\): \(x = 3\).

Step 5: Check the Solution

Plug \(x = 3\) back into the original equation to verify: Calculate \(100^3 = (10^2)^3 = 10^6\) and \(1,000^{3-1} = (10^3)^2 = 10^6\). Both sides equal \(10^6\), confirming that \(x = 3\) is correct.

Key Concepts

Exponential Functions

Exponential functions are mathematical expressions where the variable is an exponent. They are in the form of \( f(x) = a^{x} \), where \( a \) is a constant called the base and \( x \) is the exponent or power. This means the function grows by a constant multiplicative rate. Exponential functions are recognizable due to their rapid increase or decrease, depending on the base value. If the base is greater than 1, the function will grow exponentially. If the base is between 0 and 1, the function will decrease towards zero.
When dealing with equations like \( 100^x = 1,000^{x-1} \), we identify both sides as exponential expressions. Converting each part into a common base simplifies solving them.

• Recognize that 100 can be written as \( 10^2 \) and 1,000 as \( 10^3 \).
• Rewrite the equation using these powers to set a common ground for simplifying.

By transforming exponential expressions into equations with like bases, it becomes easier to solve for the variable exponent.

Properties of Exponents

Understanding the properties of exponents is crucial in manipulating and solving exponential equations. One key property used in the exercise is the power of a power rule. This rule states that \((a^m)^n = a^{m \times n}\). This allows us to simplify expressions like \((10^2)^x\) to \(10^{2x}\). Recognizing and applying these rules help in reducing complex expressions into simpler forms.
Other important properties include:

• Product of powers rule: \( a^m \times a^n = a^{m+n} \)
• Quotient of powers rule: \( \frac{a^m}{a^n} = a^{m-n} \)
• Zero exponent rule: \( a^0 = 1 \), where \( a eq 0 \)

By applying these properties, we can manipulate and solve exponential equations systematically. In the equation \(10^{2x} = 10^{3(x-1)}\), the power of a power rule helps in making both sides the same base, letting us equate the exponents directly for further solving.

Linear Equations

Linear equations are algebraic expressions involving constants and a single variable raised to the first power. They stand in the form \( ax + b = c \). Solving linear equations involves isolating the variable on one side by performing arithmetic operations.
When solving exponential equations like the one in the example, reducing them to a linear form often involves equating the exponents once the bases are the same. This step transforms it into a linear equation, such as \(2x = 3(x-1)\).
To solve, we distribute constants, collect like terms, and eventually isolate the variable:

• Distribute constants: Expand expressions, for example, \(3(x-1)\) becomes \(3x - 3\).
• Collect like terms: Arrange all variable terms on one side and constants on the other.
• Isolate the variable: Finally, divide or multiply to solve for the variable, ensuring the solution checks with the original equation.

Linear equations, although simpler, provide the foundational steps to determine the solution of the transformed exponential cases we encounter.