Problem 97 - Algebra 2

Question

Write each equation in logarithmic form. \(5^{3}=125\)

Step-by-Step Solution

Answer

The logarithmic form of the equation \(5^{3}=125\) is \(\log_{5}125 = 3\).

Step 1: Identify the Base, Exponent and Result

From the given exponential equation \(5^{3}=125\), we can identify our base \(a\) = 5, exponent \(b\) = 3 and result \(c\) = 125.

Step 2: Write the Equation in Logarithmic Form

In the logarithmic form, the base of the logarithm is the same as the base of the exponent. The number we are taking the logarithm of is the result of the equation. The value of the logarithm will be the exponent in the equation. Hence, we write the equation as: \(\log_{a}c = b\)

Step 3: Substitute Values

By substituting our identified values into logarithmic equation we get: \(\log_{5}125 = 3\)

Key Concepts

Exponential Equations

Exponential equations are equations where variables appear as exponents. These types of equations commonly take the form of \(a^b = c\). Here, \(a\) is known as the base, \(b\) is the exponent, and \(c\) is the result or outcome. An essential characteristic of these equations is their ability to grow quickly when the base is greater than 1.
Understanding exponential equations is crucial, especially in disciplines like finance, biology, and physics, where growth rates and other exponential processes frequently occur. Moreover, solving exponential equations often requires manipulation through various techniques such as logarithms, because they provide a method for handling the exponents.
It’s helpful to remember that rewriting exponential equations in a different form can simplify complex problems and allow for easier solutions.

Base of an Exponent

The base in an exponential equation is the number that is repeatedly multiplied. In the equation \(5^3 = 125\), the number 5 is the base. It represents the starting point of exponential growth.

• For instance, \(5^3\) means 5 multiplied by itself three times: 5 x 5 x 5.
• The role of the base is crucial as it determines the rate of growth or decrease in an exponential situation.

Different bases can drastically alter the outcomes of exponential equations, highlighting the importance of carefully identifying and working with the correct base in any given problem.
In the logarithmic context, the base of the exponent translates directly into the base of the logarithm. Understanding how the base functions in both exponential equations and logs is key to converting between these two forms seamlessly.

Logarithmic Equations

Logarithmic equations are equations that involve a logarithm, which is an inverse operation to exponentiation. Just as multiplication is the opposite of division, logarithms undo exponentiation. In basic terms, a logarithm is asking: "To what exponent must the base be raised, to produce this number?"
For instance, converting the exponential equation \(5^3 = 125\) to \(\log_{5}{125} = 3\) involves using the logarithm to encapsulate the question: "5 raised to what power equals 125?"
Solving logarithmic equations often involves identifying and isolating the logarithmic expression, sometimes applying logarithmic properties to reach a solution. Understanding how logarithmic equations are structured allows you to rewrite and solve complex exponential problems efficiently.

Logarithm Properties

Logarithms have several useful properties that help solve equations and simplify complex expressions. Understanding and applying these properties is crucial for working effectively with both exponential and logarithmic equations. Here’s a brief overview:

• Product Property: \(\log_b{(xy)} = \log_b{x} + \log_b{y}\). This allows the multiplication within a logarithm to be split into the sum of two logs.
• Quotient Property: \(\log_b{\left(\frac{x}{y}\right)} = \log_b{x} - \log_b{y}\). Division within the log is transformed into a subtraction of logs.
• Power Property: \(\log_b{(x^n)} = n \cdot \log_b{x}\). Allows an exponent inside the logarithm to be brought outside as a multiplier.

These properties facilitate transforming and manipulating logarithmic expressions, making them easier to solve.
So when converting exponential equations to logarithms, knowing these properties can provide shortcuts and deeper insights into the problem at hand.