Problem 61 - Algebra 2 and Trigonometry

Question

In \(57-68,\) solve each equation for the variable. $$ \log _{b} 27=3 $$

Step-by-Step Solution

Answer

The solution for the base \( b \) is 3.

Step 1: Understanding the Equation

We are given the equation \( \log_b 27 = 3 \). This equation is in logarithmic form, where \( b \) is the base, \( 27 \) is the argument, and \( 3 \) is the result of the log function. We need to convert it to exponential form to solve for \( b \).

Step 2: Converting Logarithmic to Exponential Form

In general, the logarithmic equation \( \log_b a = c \) can be transformed into the exponential form \( b^c = a \). Using this relationship, we can rewrite the given log equation \( \log_b 27 = 3 \) as \( b^3 = 27 \).

Step 3: Solving the Exponential Equation

Now, we have \( b^3 = 27 \). To solve for \( b \), we take the cube root of both sides of the equation. This gives us \( b = \sqrt[3]{27} \).

Step 4: Calculating the Cube Root

The cube root of 27 is a value that, when multiplied by itself three times, equals 27. Calculating this, we find \( b = 3 \) because \( 3 \times 3 \times 3 = 27 \).

Step 5: Confirming the Solution

To ensure correctness, substitute \( b = 3 \) back into the original equation. \( \log_3 27 = 3 \) holds because \( 3^3 = 27 \). This confirms our solution is correct.

Key Concepts

Exponential Form

When dealing with logarithmic equations, converting them into an exponential form is a powerful strategy for simplifying and solving them. The relationship between the two can be summarized as follows: If you have a logarithmic equation \(\log_b a = c\), it can be rewritten in exponential form as \(b^c = a\).

• Logarithmic Form: \(\log_b a = c\)
• Exponential Form: \(b^c = a\)

The exponential form clearly shows the base \(b\) raised to the power of \(c\) equals \(a\). This transformation is useful because it changes the problem from logarithmic terms to a power relation, which is often easier to work with, especially when solving for the base \(b\).
In our exercise, converting \(\log_b 27 = 3\) to the exponential form \(b^3 = 27\) allows us to focus on finding \(b\) that will make the equation true.

Cube Root Calculation

Finding a number whose cube equals another number involves calculating the cube root. The expression \(\sqrt[3]{a}\) denotes the cube root of \(a\). In essence, the cube root of a number \(a\) is the number \(b\) such that when you multiply \(b\) by itself three times, you obtain \(a\).

For example, with our problem:

• Given: \(b^3 = 27\)
• Calculation: \(b = \sqrt[3]{27}\)
• Determine \(b\) such that \(b \times b \times b = 27\)

The cube root of 27 is 3, because \(3 \times 3 \times 3\) equals 27. Cube roots are important when you need to "undo" a cube, such as when finding the base of a number raised to a power in exponential equations.

Base of Logarithm

In a logarithmic equation like \(\log_b a = c\), the base \(b\) is a critical component. It represents the number that is raised to the power of the logarithmic result in order to reproduce the argument. Understanding the base is crucial in transforming and solving these equations.

Logarithms and their bases often relate back to exponential functions, where:

• The base is the foundation of the exponential form.
• Knowing the base allows you to convert from a logarithmic to an exponential relationship.

In our exercise, converting \(\log_b 27 = 3\) to the exponential form \(b^3 = 27\), we solved for the base \(b\). Ultimately, finding \(b = 3\) was accomplished by recognizing that the exponential relationship "undoes" the logarithmic one, confirming the base required for the logarithmic statement.