Question

Solve each equation. $$ \log _{3} x=2 $$

Short answer

x = 9

Step-by-step solution

Understand the Logarithmic Equation

The given equation is in the form \(\text{log}_{a} b = c\), which implies that \(a^c = b\). In this problem, \( a = 3\), \( b = x\), and \( c = 2\).

Convert the Logarithmic Form to Exponential Form

Rewrite the logarithmic equation \(\text{log}_{3} x = 2\) in its exponential form. This gives \(3^2 = x\).

Solve the Exponential Equation

Calculate \(3^2\). \(3^2 = 9\). Hence, \(x = 9\).

Key concepts

logarithmic functions

Logarithmic functions are essential in mathematics as they help us reverse the process of exponentiation. If you know the value of an exponent and the base, a logarithmic function can find the original value. For example, in the logarithmic equation \(\log_{3} x = 2\), we aim to find the value of \{x}\ such that when 3 is raised to that power, we get 2. The general form is \(\log_{a} b = c\), which translates to \{a^c = b}\. Here, \{a}\ is the base, \{b}\ is the result, and \{c}\ is the exponent. This concept is invaluable across various fields including science and finance.Continuously practicing the conversion between log and exponential forms will boost your confidence and accuracy in solving equations.

exponential form

Understanding the exponential form is key to solving logarithmic equations. The exponential form can be derived from the logarithmic form. In our example \(\log_{3} x=2\), converting this to exponential form means rewriting the equation as \(3^2 = x\). This tells us that 3 raised to the power of 2 equals \{x}\. Thus, we have eliminated the logarithm and now only need to compute the exponent. Let's calculate \{3^2}\. Since \{3^2 = 9}\, we can conclude that \{x = 9}\. This step is crucial because the exponential form visually shows the relationship between the variables, making it easier to solve the equation.

solving equations

Solving equations involving logarithms is a multi-step process. Here’s the breakdown for our example: First, identify the given logarithmic equation and understand its form. For \(\log_{3} x = 2\), we recognize it follows the general form \{\log_{a} b = c}\. Next, convert this logarithmic form to its exponential counterpart. This step simplifies the equation. So, \(\log_{3} x = 2\) becomes \{3^2 = x}\. Finally, solve the exponential equation by calculating the power. Since \{3^2 = 9}\, we have \{x = 9}\. Following these steps systematically ensures the solution is correct and helps reinforce your understanding of the underlying principles. Make sure to practice regularly as it helps develop the skills to read and manipulate equations effortlessly.