Problem 25 - Intermediate Algebra

Question

For Problems \(1-34\), solve each equation. $$ 10^{x}=0.1 $$

Step-by-Step Solution

Answer

The solution is \( x = -1 \).

Step 1: Rewrite the Equation Using Powers of 10

We know that 0.1 can be expressed as a power of 10, specifically, 0.1 is the same as \( 10^{-1} \). So, rewrite the equation as \( 10^x = 10^{-1} \).

Step 2: Set the Exponents Equal

Since the bases on both sides of the equation are the same (both are base 10), we can set the exponents equal to each other. Thus, \( x = -1 \).

Step 3: Verify the Solution

Substitute \( x = -1 \) back into the original equation to verify: \( 10^{-1} = 0.1 \). This is correct, so \( x = -1 \) is indeed the solution.

Key Concepts

Understanding Powers of Ten

The term "powers of ten" involves expressing numbers as exponents of base 10. This system is widely used because our numerical system is base 10, which means each digit represents a power of ten. For example:

• 10⁰ = 1
• 10¹ = 10
• 10² = 100
• 10⁻¹ = 0.1

When we say a number like 0.1 is equal to 10⁻¹, it means that we're taking 1 divided one time by 10. This is handy for transforming decimals into more manageable forms using exponents.
If you grasp this concept, it simplifies solving exponential problems by aligning your values in the same format: as powers of ten.

The Process of Solving Exponential Equations

Solving exponential equations often involves a mix of algebraic techniques and a firm understanding of exponents. The general idea is to isolate the variable on one side of the equation. In the context of base 10, this means rewriting each side of the equation to have the same base before comparing their exponents. Here's a simple approach:

• Express both sides of the equation as powers of 10.
• If the bases are the same, set the exponents equal to each other.
• Solve for the variable.

This method hinges on the property that, if two numbers with the same base are equal, their exponents must also be equal. If you're solving an equation like 10^x = 0.1, rewrite 0.1 as 10⁻¹, set x = -1 by equating exponents, then check your solution by substitution. This will ensure correctness and reinforce your understanding.

Introduction to Logarithms

Logarithms are closely tied to exponents and provide an excellent tool for solving equations where the variable is an exponent. Understanding logarithms starts with the definition: a logarithm is the inverse operation of exponentiation. This means:

• If 10^x = y, then log₁₀(y) = x.

Logarithms convert multiplicative processes into additive ones. Therefore, when we encounter an exponential equation we're unsure of, we may utilize logarithms to solve for the exponent. To solve 10^x = 0.1 using logarithms, we take the logarithm base 10 of both sides: log₁₀(10^x) = log₁₀(0.1), which simplifies to x = -1, confirming the initial solution. Using logarithms is efficient, especially with more complicated equations, and provides deeper insight into the relationships between different mathematical operations.