Problem 4 - Algebra and Trigonometry Real Mathematics, Real People

Question

Can you solve \(5^{x}=125\) using a One-to-One Property?

Step-by-Step Solution

Answer

So, solving \(5^{x} = 125\) using the One-to-One Property gives the result \(x = 3\).

Step 1: Expressing 125 as a power of 5

First, notice that you can also write 125 as 5 to the power of 3, i.e. \(5^3 = 125\). Therefore, the equation can be rewritten as \(5^x = 5^3\).

Step 2: Applying the One-to-One property

Now we use the One-to-One property of exponents, which says that, for any real numbers `m`, `n`, and any positive number `b` (b ≠ 1), if `b^m = b^n`, then `m = n`. So, from the equation \(5^x = 5^3\), we can say `x = 3`.

Key Concepts

Exponential Equations

Exponential equations are equations in which variables appear as exponents. A classic form of an exponential equation is something like \( b^x = N \), where \( b \) is the base and \( x \) is the exponent. Solving these types of equations often involves expressing both sides of the equation with the same base, making it easier to compare the exponents.

Consider the problem \( 5^x = 125 \). Here, the key is recognizing that both 5 and 125 can share the same base: 5. By converting 125 into \( 5^3 \), we transform the original equation into an equation where both sides have the same base. This allows us to utilize the One-to-One Property of exponents to find the value of \( x \).

Understanding exponential equations is crucial because they frequently appear in various scenarios involving exponential growth and decay, such as in physics, finance, and biology. Recognizing and solving them effectively relies on manipulating the exponents and bases accurately.

Properties of Exponents

To solve exponential equations, one must be comfortable with the properties of exponents. These rules enable you to manipulate and simplify expressions involving powers. Here are some essential properties:

• Product of Powers: \( b^m \times b^n = b^{m+n} \)
• Power of a Power: \( (b^m)^n = b^{m \cdot n} \)
• Quotient of Powers: \( \frac{b^m}{b^n} = b^{m-n} \) when \( b eq 0 \)

These properties allow us to rewrite and solve exponential equations by expressing numbers as powers of a common base.

For instance, in the equation \( 5^x = 125 \), you use the fact that \( 125 = 5^3 \). Familiarity with these concepts is integral to solving similar problems swiftly and intuitively throughout algebra.

Algebra Problem Solving

Problem-solving in algebra involves not just knowing the rules, but strategically applying them to reach a solution. This requires a blend of observation, pattern recognition, and logical reasoning.

When approaching an algebra problem like \( 5^x = 125 \), you begin by simplifying and expressing terms in a compatible form, enabling the application of algebraic properties efficiently.

The One-to-One Property of exponents is key in problems where both sides of an equation can be rewritten with the same base. Once similar bases are established, the logic follows easily because you deduce that the exponents have to equate, leading directly to a solution.

Moreover, practicing these problem-solving strategies enhances your skill in tackling more complex algebraic equations, allowing you to handle all types of algebraic phenomena effectively.