Problem 53 - Precalculus : Building Concepts and Connections

Question

Use the definition of a logarithm to solve for \(x\). $$\log _{3} x=\frac{1}{3}$$

Step-by-Step Solution

Answer

The solution for \(x\) in the equation \(\log _{3} x=\frac{1}{3}\) is \(1.442\), rounded to three decimal places.

Step 1: Convert Logarithmic to Exponential Form

The logarithmic form of the equation given is \(\log _{3} x=\frac{1}{3}\). In order to find \(x\) it helps to convert this into an exponential form using the rule \(a = \log_b (c)\), which translates to \(b^a = c\). Here,\(b = 3\), \(a = \frac{1}{3}\), and \(c = x\). Applying the rule, we can rewrite the equation as \(3^{\frac{1}{3}} = x\).

Step 2: Evaluate the Exponential Expression

Next, it's necessary to calculate the value of the exponential expression \(3^{\frac{1}{3}}\) to find \(x\). The expression \(3^{\frac{1}{3}}\) means the cube root of 3. After calculating, we find that \(x = 3^{\frac{1}{3}}\) is approximately \(1.442\).

Key Concepts

Exponential Form

In mathematics, when we talk about the exponential form, we are referring to an expression where a number is expressed as a base raised to some power. This is highly useful for solving equations, particularly those that involve logarithms. The exponential form is often used as a bridge to translate between different mathematical expressions and make them easier to solve.

For example, with a logarithmic equation such as \( \log_b (c) = a \), we can switch to its exponential form \( b^a = c \). This means the base \( b \) is raised to the power of \( a \), resulting in the value \( c \). This transformation is crucial when you want to solve for an unknown in logarithmic equations because it clearly presents the relationship between the base and the resulting value.

In our exercise, the log equation \( \log_3 x = \frac{1}{3} \) converts to \( 3^{\frac{1}{3}} = x \) by using this principle, making it ready for direct computation to find \( x \). Thus, understanding exponential form is a key step when solving logarithmic equations.

Cube Root

The cube root is a fundamental concept in mathematics, particularly when dealing with powers and roots in algebra. It is the operation that reverses cubing a number. For instance, if the number \( y \) is cubed, then the cube root of \( y^3 \) is \( y \).

Mathematically, the cube root of a number \( n \) is expressed as \( n^{\frac{1}{3}} \). This expression is the inverse of raising a number to the third power. The cube root helps us to simplify expressions and solve equations that involve cubic powers.

• The cube root of 27 is 3, because \( 3^3 = 27 \).
• Similarly, the cube root of 8 is 2, because \( 2^3 = 8 \).

In our exercise, when we find \( x = 3^{\frac{1}{3}} \), it means we are calculating the cube root of 3, which gives us an approximate value of \( 1.442 \). This shows how the cube root is applied in practical logarithmic and exponential problems.

Logarithmic Equation

A logarithmic equation involves a logarithm of a variable and is an essential concept in understanding exponential growth, decay processes, and many areas of mathematics and science. It helps express exponential relationships in a linear form, making complex calculations more manageable.

The general form of a logarithmic equation \( a = \log_b (c) \) indicates that there exists a value \( c \) that is the result of raising the base \( b \) to the power \( a \). Logarithmic equations are often solved by changing them into their corresponding exponential forms, as it simplifies finding unknown variables.

• The base, \( b \), must be a positive number not equal to 1.
• The value \( a \) represents the exponent that the base \( b \) is raised to in order to get \( c \).
• For example, in \( \log_2 8 = 3 \), the base \( 2 \) raised to the power of \( 3 \) results in \( 8 \).

Understanding how to manipulate logarithmic equations is a fundamental skill required for advanced mathematical problem-solving, as demonstrated in our exercise when solving for \( x \) in \( \log_3 x = \frac{1}{3} \).