Question

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

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

Algebra 2 and Trigonometry

Algebra 2 and Trigonometry

Ann Xavier Gantert 1st Edition

Chapter 7, Problem 27

Instant Answer

Solved by Expert Heather Zimmers

04/30/2020

Step 1

We know that $100 = 10^{2}$ and $1,000 = 10^{3}$. So, the equation becomes $(10^{2})^{x} = (10^{3})^{x-1}$. Show more…

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

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Key Concepts

Properties of Exponents

The laws of exponents, such as a^(m+n) = a^m * a^n and (a^m)^n = a^(mn), enable the manipulation and simplification of expressions involving powers. These rules are central to rewriting and solving equations that involve exponentials.

Base Conversion

This concept involves rewriting numbers as powers of a common base to simplify comparisons between exponential expressions. By expressing different terms using the same foundational number, it becomes easier to equate the exponents and solve the equation.

One-to-One Property of Exponential Functions

When dealing with exponential functions with the same base (provided the base is positive and not equal to 1), the one-to-one property states that if a^m = a^n then m = n. This allows the original exponential equation to be converted into a simpler algebraic equation.

Verification of Solutions

After solving an equation, checking the solution by substituting it back into the original equation ensures the accuracy and validity of the result. This step is crucial in confirming that all conditions of the original problem are met.

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