Problem 28 - College Algebra with Corequisite Support

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: Identify the Form of the Logarithm

The given equation is a logarithm: \( \log_{5}(x) = 2 \). This equation denotes that 5 raised to some power equals \( x \).

Step 2: Rewrite in Exponential Form

Convert the logarithmic equation into its exponential form. The logarithmic equation \( \log_{b}(a) = c \) can be rewritten as \( b^c = a \). In this exercise, it becomes \( 5^2 = x \).

Step 3: Calculate the Exponent

Evaluate the exponential expression. Here, calculate \( 5^2 \), which equals 25. So, \( x = 25 \).

Key Concepts

Logarithmic Equation

A logarithmic equation is an expression that uses logarithms to relate quantities. It typically takes the form \( \log_b(a) = c \), meaning the logarithm of \(a\) with base \(b\) is equal to \(c\). In simpler terms, it answers the question: "To what power must the base \(b\) be raised to obtain \(a\)?"
For instance, in the equation \( \log_{5}(x) = 2 \), we are looking for which power 5 must be raised to, in order to get \(x\). Notice how the base here is 5, the value inside the log is \(x\), and the log is equal to 2. Understanding this setup is crucial, as it is the gateway to rewriting things in exponential form, which can simplify solving for \(x\).

Base Conversion

Base conversion involves changing the format of a given problem by using different bases. In the context of logarithmic and exponential forms, it refers to converting a logarithmic equation into exponential form or vice versa. This is a snapshot of mathematical translation skills.
When you encounter a logarithmic equation like \( \log_b(a) = c \), converting this into an exponential format looks like \( b^c = a \). The base \(b\) remains unchanged; however, understanding how these conversions work allows you to see the same relationship from a different perspective. This technique is particularly useful when solving logarithmic equations as it often simplifies the equation to basic arithmetic.

Solving Equations

Solving equations is a fundamental part of algebra that focuses on finding the values of unknown variables. To solve a logarithmic equation typically involves restructuring it to remove the log function, often by converting it to an exponential form. This method opens a straightforward path to isolating and solving for the variable.
By converting the equation \( \log_{5}(x) = 2 \) to the exponential form \( 5^2 = x \), we transform the problem into a simpler arithmetic calculation. Rather than working through the complexities of logarithms, the equation becomes one where you simply need to square the base number to find \(x\) – in this instance, \(5^2\) results in 25, thus solving for \(x\).

Algebraic Manipulation

Algebraic manipulation refers to rearranging and rewriting equations to make them easier to solve. This may involve actions like expanding expressions, factoring, or converting forms such as reversing logs to exponentials, as seen in logarithmic equations.
For example, the given equation \( \log_{5}(x) = 2 \) was manipulated by converting it into exponential form, \( 5^2 = x \). This transition is a subtle yet powerful example of algebraic manipulation as it drastically simplifies the equation, leading to an immediate solution. Mastering such techniques equips you with tools to tackle more complicated problems down the road, as you learn to see the connections between different mathematical representations and operations.