Problem 28 - College Algebra

Question

For the following exercises, solve for \(x\) by converting the logarithmic equation to exponential form. $$\log _{5}(x)=2$$

Step-by-Step Solution

Answer

\(x = 25\)

Step 1: Understand the Logarithmic Equation

The given equation is \(\log _{5}(x) = 2\). This means that the base is 5, and we are trying to find what power we need to raise 5 to in order to get \(x\).

Step 2: Convert to Exponential Form

To convert the logarithmic equation to an exponential equation, use the definition of a logarithm: \(\log_b(a) = c \Rightarrow b^c = a\). Applying this, we have \(5^2 = x\).

Step 3: Solve for x

Now, compute the value of \(5^2\): \[5^2 = 25\]. Thus, \(x = 25\).

Key Concepts

Logarithmic Equations

Logarithmic equations involve the use of logarithms. Think of a logarithm as the reverse of exponents. In a logarithm, you need to determine what power a number must be raised to, to produce another number. For instance, in the equation \(\log_{5}(x) = 2\), the base 5 is raised to what power to result in \(x\)? This is a typical logarithmic equation where

• The base is 5,
• The value portion is \(x\), and
• The logarithmic result is 2.

Understanding how each component works will make converting to exponential form much easier. It is a foundational step in solving the equation for \(x\).

Solving for x

Solving for \(x\) in equations involves finding out the value that satisfies the equation. Once a logarithmic equation is transformed into exponential form, finding \(x\) becomes straightforward. After conversion, the equation is written in terms of powers:

• The base from the logarithm becomes the base of the exponent,
• The logarithmic result becomes the power,
• The value portion becomes the result of the power operation.

In the example \(\log_{5}(x) = 2\), solving for \(x\) means changing the expression to \(5^2 = x\). This gives you a direct computation, \(x = 25\). With these simple calculations, you can quickly determine \(x\).

Converting Forms

Converting forms is crucial in algebra, especially when solving logarithmic equations. Here, it involves translating a logarithmic equation into an exponential one, allowing you to leverage exponential rules to find solutions. The conversion formula \[\log_b(a) = c \Rightarrow b^c = a\]serves as an essential tool in this process. Applying this formula, you can reframe any equation: for instance, \(\log_{5}(x) = 2\) becomes \(5^2 = x\). Conversion changes the problem’s appearance to reach a more familiar and solvable format. Practicing these conversions enhances algebraic intuition and problem-solving skills. Remember, once converted, solving the equation becomes straightforward. Converting forms is like translating a problem into a language you more fully understand, simplifying the process to solve it.