Problem 11 - College Algebra

Question

For the following exercises, use logarithms to solve. $$ 9^{x-10}=1 $$

Step-by-Step Solution

Answer

x = 10

Step 1: Understanding the Equation

The given equation is \( 9^{x-10} = 1 \). We need to solve for \( x \) using logarithms or properties of exponents.

Step 2: Recall Power of 1

Recall that any number raised to the power of 0 is 1, i.e., for any base \( a \), if \( a^0 = 1 \), then the exponent must be 0.

Step 3: Equate the Exponent to Zero

Since the left side of the equation is \( 9^{x-10} \), for it to equal 1, the exponent must be zero. Thus, set \( x-10 = 0 \).

Step 4: Solve for x

Solve the equation \( x - 10 = 0 \) by adding 10 to both sides to get \( x = 10 \).

Step 5: Verify the Solution

Substitute \( x = 10 \) back into the original equation: \( 9^{10-10} = 9^0 = 1 \), confirming that left side equals the right side.

Key Concepts

Exponent Properties

Understanding exponent properties is essential when dealing with logarithmic equations like the given problem. An exponent tells us how many times to multiply a number, known as the base, by itself. For instance, in the term \( 9^3 \), 9 is the base and 3 is the exponent, which means you multiply 9 by itself three times: \(9 \times 9 \times 9\).

One crucial exponent property used in the original exercise is the "Power of Zero" rule. This states that any non-zero number raised to the power of zero is equal to 1. That is, for any base \( a \), \( a^0 = 1 \).

Using this property, it's clear that for the expression \( 9^{x-10} \to 1 \), the exponent \( x-10 \) must equal zero, because this will make \( 9^0 = 1 \). Such knowledge about properties of exponents significantly simplifies solving equations.

Solving Equations

Solving equations involves finding a value for a variable that makes the equation true. In our case, the equation is \( 9^{x-10} = 1 \).

We start by applying the properties of exponents, as discussed earlier. Recognizing that \( a^0 = 1 \) for any non-zero \( a \), we equate the exponent of our given equation to zero: \( x-10 = 0 \).

This kind of simplification transforms the problem into a straightforward algebraic equation. By solving \( x-10 = 0 \), we isolate \( x \) to find its value. To achieve this, we add 10 to both sides of the equation, resulting in \( x = 10 \).

Such systematic approaches to isolating and solving for variables are key approaches in algebra, making complex-looking problems more approachable.

Verification in Mathematics

Verification ensures that our solution to an equation is correct. It's a critical step in solving equations as it confirms the validity of our result. For this problem, verifying the solution \( x = 10 \) involves substituting it back into the original equation: \( 9^{x-10} = 1 \).

By substituting, we perform the operation: \( 9^{10-10} = 9^0 \). Given the exponent property \( 9^0 = 1 \), our calculation checks out since both sides of the equation equal 1.

Verification helps spot errors by enabling a reassessment. Without this vital step, we might proceed with incorrect solutions inadvertently. Verification is a shield against simple algebraic mistakes and promotes confidence in our problem-solving process.