Problem 22 - A Graphical Approach to College Algebra

Question

Solve each equation. Give the exact answer. $$\log _{x} 5^{2}=6$$

Step-by-Step Solution

Answer

The solution is \(x = 5^{1/3}\).

Step 1: Understand the Given Equation

The given equation is \( \log_{x} 5^2 = 6 \). This means that the logarithm of \(5^2\) with base \(x\) is equal to 6. Our goal is to find the base \(x\).

Step 2: Apply the Definition of Logarithms

By definition, if \( \log_{x} a = b \), then \( x^b = a \). In our equation, \(a = 5^2\) and \(b = 6\). Thus, we have the equation \( x^6 = 5^2 \).

Step 3: Simplify \(5^2\)

Calculate \(5^2\), which is equal to 25. So, the equation \( x^6 = 5^2 \) becomes \( x^6 = 25 \).

Step 4: Solve for \(x\) Using the Sixth Root

To isolate \(x\), take the sixth root of both sides of the equation: \(x = 25^{1/6}\).

Step 5: Write the Exact Solution

The exact solution is \( x = 5^{2/6} \), which simplifies to \( x = 5^{1/3} \).

Key Concepts

Base of Logarithm

A logarithm has a base that forms the foundation of calculations. The base indicates the number raised to a particular power to produce a given number.
In the example, we are working with

• A logarithmic expression: \( \log_{x} 5^2 \)
• Where \( x \) is the unknown base of the logarithm
• The goal is to find this base.

The base in a logarithm tells us how many times we multiply the base number by itself to reach another number.
Understanding the base is crucial because it converts a complex logarithmic expression into a simpler form to solve further. For instance, \( \log_{x} a = b \) implies that \( x^b = a \), which is key to solving such equations. By setting the base correctly, we can transform logarithmic equations into exponential ones that are easier to handle.
This becomes particularly useful when dealing with expressions like \( \log_{x} 5^2 \), as identifying the base allows us to convert the problem into an exponential format for straightforward resolution.

Exponential Equation

An exponential equation involves a variable appearing in the exponent. These equations often appear when dealing with logarithms. For our logarithmic problem, the equation:

• unravels into an exponential format through \( x^6 = 25 \)

using the properties of logarithms.
Recognizing the transformation into an exponential equation is essential.
This equation tells us that number \( x \), raised to the power of 6, equals 25. Solving exponential equations involves isolating the base and finding the root of both sides. For instance, using the equation \( x^6 = 25 \), we take the sixth root of both sides to isolate \( x \).
Therefore, the understanding of transforming logarithmic statements into exponentials allows us to use different mathematical tools and methods to reach the solution.

Properties of Exponents

The properties of exponents are mathematical rules that simplify expressions and help solve equations.
They allow us to handle powers more effectively through operations like multiplication and division of powers.
Key exponent properties include:

• Product of powers: \( a^m \times a^n = a^{m+n} \)
• Power of a power: \( (a^m)^n = a^{mn} \)
• Power of a product: \( (ab)^m = a^m b^m \)
• Root as fractional power: \( a^{1/n} = \sqrt[n]{a} \)

In solving the original exercise, these properties came into play, especially the root as a fractional power.
In our example, after forming the exponential equation \( x^6 = 25 \), we used the property of roots as fractional powers to simplify and find the solution: \( x = 25^{1/6} \). By understanding these properties, we can simplify complex expressions into manageable calculations, bringing clarity and precision to our work.