Problem 24 - Algebra 2 and Trigonometry

Question

In \(15-26,\) write each logarithmic equation in exponential form. $$ \log _{8} 2=\frac{1}{3} $$

Step-by-Step Solution

Answer

In exponential form, it is \(8^{\frac{1}{3}} = 2\).

Step 1: Identify the Base, Result, and Exponent

In the given logarithmic equation, \( \log_{8} 2 = \frac{1}{3} \), the base is \(8\), the result is \(2\), and the exponent is \(\frac{1}{3}\). This means that 8 raised to the power of \(\frac{1}{3}\) equals 2.

Step 2: Write the Exponential Form

Using the identified components, we rewrite the logarithmic equation in exponential form as follows:\[ 8^{\frac{1}{3}} = 2 \]

Step 3: Recheck the Exponential Form

Verify the exponential equation by computing \(8^{\frac{1}{3}}\). Since \(8 = 2^3\), we have:\[ (2^3)^{\frac{1}{3}} = 2^{3 \cdot \frac{1}{3}} = 2^1 = 2 \]This confirms the exponential form is correct.

Key Concepts

Exponential Form

Understanding exponential form is essential when working with logarithmic equations. An exponential form presents a relationship between a base, an exponent, and a result. It is expressed as \( b^E = R \), where \(b\) is the base, \(E\) is the exponent, and \(R\) is the result obtained by raising the base to the power of the exponent.
For example, in our exercise, the exponential form is \(8^{\frac{1}{3}} = 2\). This shows how the number 8, when raised to the exponent \(\frac{1}{3}\), results in the number 2. Transforming logarithmic equations into exponential form is key to simplifying expressions and solving for unknowns, especially in algebra, calculus, and various applications in science.

Base and Exponent Identification

To convert a logarithmic equation to its exponential form, identifying the base, exponent, and result is crucial. In a logarithmic expression \(\log_b R = E\), the base \(b\) is the number that is being raised to a power, the exponent \(E\) is the power to which the base is raised, and the result \(R\) is the number we obtain.
In the problem at hand, the base is \(8\), the exponent is \(\frac{1}{3}\), and the result is \(2\). This step is foundational because it ensures that we correctly rewrite the expression in exponential form. The ability to accurately identify these components aids understanding the essence of logarithms, which are essentially exponents in disguise, retrieving a specific result when a base is exponentially manipulated.

Logarithmic to Exponential Conversion

Converting from logarithmic to exponential form is a pivotal skill, making it simpler to handle equations involving exponents. This conversion uses the correspondence between the logarithmic form \(\log_b R = E\) and the exponential form \(b^E = R\).
In our example, \(\log_{8} 2 = \frac{1}{3}\) converts to \(8^{\frac{1}{3}} = 2\). To verify, compute \(8^{\frac{1}{3}}\). Since \(8 = 2^3\), this becomes \((2^3)^{\frac{1}{3}} = 2^{3\times \frac{1}{3}} = 2^1 = 2\), confirming our conversion. Mastering this skill simplifies logarithmic expressions and facilitates seamless expertise across various mathematical and scientific fields, where such equations frequently arise.