Problem 13 - College Algebra

Question

Write each equation in its equivalent logarithmic form. $$ \sqrt[3]{8}=2 $$

Step-by-Step Solution

Answer

\(\log_{8}{2} = \frac{1}{3}\).

Step 1: Identify the base, exponent and result

In the equation \(\sqrt[3]{8}=2\), the number \(8\) is the base, \(2\) is the result and the fractional exponent \(3^{-1}\) can be seen as the logarithm of the result based on the base. Hence, \(8\) is the base, \(3^{-1}\) is the exponent and \(2\) is the result.

Step 2: Convert to Logarithmic Form

To convert this into logarithmic form, it is written as \(\log_{base}{result} = exponent\). Substituting our identified base, exponent and result, return \(\log_{8}{2} = \frac{1}{3}\). This reads as 'log base 8 of 2 equals to 1 over 3'.

Key Concepts

Logarithms

Logarithms are like detectives in the world of mathematics, solving the mystery of what exponent is needed to produce a certain number when applied to a base. The relationship between logarithms and exponents is an inverse one; while an exponent tells us the power we raise a base to get a certain number, a logarithm tells us what power the base was raised to reach that number. In other words, if we know the base and the result, a logarithm will tell us the exponent.

Taking the exercise of converting \(\sqrt[3]{8}=2\) into logarithmic form as an example, we treat the cube root of 8 as a radical expression, where 8 is the result of raising the base (2) to the power of the exponent (3). To find the logarithmic form, we ask ourselves, 'What power must we raise 8 to, in order to get 2?' The logarithmic form, \(\log_{8}{2} = \frac{1}{3}\), answers this question succinctly, telling us that 8 raised to the power of 1/3 gives us 2.

Exponents

Exponents come into play when we repeatedly multiply a number by itself. Think of it as a shortcut to writing a long multiplication. For example, \(2^3\) tells us to multiply 2 by itself three times: 2 x 2 x 2, which equals 8. At times, we even encounter fractional exponents, like the one in our exercise \(\sqrt[3]{8}\), which is equivalent to \(8^{1/3}\). This tells us that we're looking for the number that, when raised to the power of 3, results in 8.

In this light, an exponent serves as a clear and concise means to express complex multiplications. It's essential to realize that every radical expression has an equivalent expression with a fractional exponent, which simplifies understanding and solving radical equations.

Radicals

Radicals express the 'root' operation, which basically asks the question: 'Which number, when raised to a certain power, gives me the number under the radical sign?' The cube root, square root, fourth root, and so on, all fall under this category.

A radical can also be expressed as a fractional exponent, with the denominator representing the root's degree. For instance, the cube root of 8 \(\sqrt[3]{8}\), can be expressed as \(8^{1/3}\). This relationship with exponents simplifies operations, particularly when dealing with higher-level algebra and calculus. Understanding how to express radicals in exponential form and vice versa is a crucial skill in mathematics that can make solving equations much more manageable.