Problem 54 - Intermediate Algebra

Question

Write each logarithm without an exponent or a radical symbol. Then simplify, if possible. See Example \(5 .\) $$ \log \sqrt[3]{7} $$

Step-by-Step Solution

Answer

\(\frac{1}{3} \log(7)\)

Step 1: Identify the Radical

The term given is \(log \sqrt[3]{7}\). The radical sign indicates that 7 is under a cube root, meaning it can be rewritten as an exponent.

Step 2: Rewrite the Radical using Exponents

A cube root can be expressed as a power of \(\frac{1}{3}\). Therefore, \(\sqrt[3]{7}\) can be written as \(7^{\frac{1}{3}}\).

Step 3: Apply the Logarithm Power Rule

The power rule of logarithms states that \(\log(a^b) = b \times \log(a)\). By applying this rule, the expression becomes \(\frac{1}{3} \times \log(7)\).

Step 4: Simplify the Expression

Since \(\log(7)\) cannot be simplified further without specific logarithm tables, the final simplified expression is \(\frac{1}{3} \log(7)\).

Key Concepts

Radicals

Radicals are symbols that denote roots of numbers. They often have a small number written in the "crook" of the symbol, which indicates the degree of the root. For example,

• The square root is denoted by \(\sqrt{}\) and represents a root of degree 2.
• The cube root, used in this exercise, is denoted by \(\sqrt[3]{}\) and represents a root of degree 3.

To express radicals in a different way, we use exponents. The cube root of 7, or \(\sqrt[3]{7}\), is equal to \(7^{\frac{1}{3}}\). This method of conversion helps in easily applying properties of exponents and logarithms later on. Remember, each radical can be expressed as an exponent by using a reciprocal power. This is a crucial concept because handling radicals in this format allows for straightforward calculations via algebraic rules.

Exponents

Exponents are a way to represent repeated multiplication. They tell us how many times to multiply a number by itself. The expression \(a^n\) indicates "a to the power of n," meaning that "a" is multiplied by itself "n" times. When it comes to radicals and their conversion:

• A square root can be written as \(a^{\frac{1}{2}}\).
• A cube root can be written as \(a^{\frac{1}{3}}\).

This transformation is vital for simplifying expressions involving logarithms. By using powers, you can apply logarithmic identities more easily, which leads us directly into the next topic: the logarithm power rule.

Logarithm Power Rule

The logarithm power rule is a handy tool that helps simplify expressions involving powers within a logarithm. The rule states:\[ \log(a^b) = b \times \log(a) \]This means that when you have a logarithm of a number raised to a power, you can "take down" the exponent and multiply it by the logarithm of the base number itself. For example, as in our original problem with the cube root of 7, when expressed as \(7^{\frac{1}{3}}\), the logarithm power rule gives us:

• \(\log(7^{\frac{1}{3}}) = \frac{1}{3} \times \log(7)\)

This is particularly useful as it helps break down complex expressions into more manageable parts, making it much easier to perform further calculations or simplifications. Understanding and using these rules allows for efficient manipulation of logarithmic expressions.