Problem 29 - College Algebra

Question

For the following exercises, solve for \(x\) by converting the logarithmic equation to exponential form. $$\log _{3}(x)=3$$

Step-by-Step Solution

Answer

The solution is \( x = 27 \).

Step 1: Understand the Logarithmic Equation

The given equation is \( \log_{3}(x) = 3 \). This logarithmic form states that 3 is the power to which the base 3 has to be raised to obtain \( x \).

Step 2: Convert to Exponential Form

To convert to exponential form, we take the base of the logarithm, which is 3, and raise it to the power on the other side of the equation, which is 3. This gives the expression \( x = 3^3 \).

Step 3: Calculate the Exponential Expression

Now, compute \( 3^3 \). This means multiply 3 by itself three times. \( 3 \times 3 = 9 \), then \( 9 \times 3 = 27 \). So, \( 3^3 = 27 \).

Step 4: Write the Solution for x

After calculating the exponential expression, we find that \( x = 27 \). This is the solution to the equation \( \log_{3}(x) = 3 \).

Key Concepts

Logarithmic Equations

Logarithmic equations are a type of equation that involve the logarithm function, which is the inverse of the exponential function. In a logarithmic equation such as \( \log_b(x) = y \), the expression tells us that \( b^y = x \). The base \( b \) is the number that gets repeatedly multiplied by itself a certain number of times, indicated by the power \( y \), to get the value \( x \).

Students often encounter logarithmic equations while solving problems that require determining unknown exponents when the base and result are given. Practicing with logarithmic equations is valuable because they frequently appear in various branches of mathematics, including algebra and calculus.

• Logarithmic equations can be converted to exponential form to simplify solving them.
• Understanding the relationship between logarithms and exponents is crucial.
• Mastery of these equations enhances your comprehension of exponential growth and decay.

Grasping logarithmic equations involves recognizing the base of the logarithm and the result it produces when raised to the given power. This knowledge aids in effectively converting and solving these equations.

Converting Logarithms to Exponents

Converting logarithms to exponents is a key step in solving logarithmic equations. This process involves transforming the logarithmic equation into an exponential form, which is often easier to manage and solve.

Take the equation \( \log_b(x) = y \). Here, the base \( b \) raised to the power \( y \) equals \( x \) in exponential form, written as \( b^y = x \). This conversion is useful because it allows you to resolve the equation by exploring the properties of exponents rather than logarithms.

• The conversion is straightforward: identify the base and the exponent from the logarithmic equation.
• Write the equation in equivalent exponential form.
• Solving the exponential equation is often simpler and faster.

For the given exercise \( \log_3(x) = 3 \), conversion gives \( x = 3^3 \). Converting allows you to directly compute exponential expressions, simplifying the path to finding solutions for \( x \).

Exponential Expressions

Exponential expressions are mathematical expressions involving a base raised to a certain power or exponent. When given an exponential expression in solving equations, such as \( b^y \), the base \( b \) is multiplied by itself \( y \) times.

The ability to confidently compute values for exponential expressions is crucial, particularly in mathematics fields focusing on growth patterns, data analysis, and scientific calculations.

• Exponents signal repeated multiplications, such as \( 3^3 = 3 \times 3 \times 3 \).
• Understanding how to compute these values aids in various mathematical applications.
• Physically, exponential expressions often deal with significant growth or decay, making them vital in real-world science and finance.

In our practice problem, after converting the logarithmic equation to \( x = 3^3 \), calculating the exponential expression gives \( x = 27 \). This value demonstrates how exponential expression evaluations determine the solution to an equation.
Exploring exponential expressions helps build a foundation for handling complex mathematics in algebra, ensuring students are equipped to solve real-life problems involving exponential growth and decay.