Problem 259 - Calculus Volume 1

Question

For the following exercises, write the equation in equivalent logarithmic form. $$ \sqrt[3]{64}=4 $$

Step-by-Step Solution

Answer

\(\log_{64}(4) = \frac{1}{3}\).

Step 1: Understand the Form of the Expression

The given expression is in the form of a root equation, which can be expressed in exponential form as \( 64^{1/3} = 4 \).

Step 2: Identify Components for Logarithm

To rewrite in logarithmic form, identify the base, the exponent, and the result. Here, \( 64 \) is the base, \( 4 \) is the result, and \( 1/3 \) is the exponent of the power that \( 64 \) is raised to in order to get \( 4 \).

Step 3: Convert to Logarithmic Form

The equivalent logarithmic expression is \( \log_{64}(4) = \frac{1}{3} \). This means that \( 64 \) raised to the power of \( \frac{1}{3} \) equals \( 4 \).

Key Concepts

Exponential Form

When we talk about exponential form, we are referring to expressing numbers using powers or exponents. Consider the example \(64^{1/3} = 4\). Here, 64 is the base number, and \(1/3\) is the exponent. The exponent tells us how many times to multiply the base by itself. However, with fractional exponents like \(1/3\), it means we take the cube root of the base.

This idea is crucial in algebra because it allows for a more compact way of writing repeated multiplication. It also plays a critical role when converting expressions to other mathematical forms, such as logarithms.

• Base: The number that gets multiplied.
• Exponent: Indicates how many times the base is used in a multiplication.
• Result: The answer we get after applying the exponent to the base.

Understanding exponential form is the stepping stone toward mastering the concept of logarithms.

Logarithms

Logarithms are the inverse of exponentiation, which means they help us find the exponent needed to obtain a certain number from a given base. In the problem \(\sqrt[3]{64} = 4\), we can express this in logarithmic form as \(\log_{64}(4) = \frac{1}{3}\).

This expression tells us that 64 raised to the power of \(\frac{1}{3}\) gives us 4. Logarithms are incredibly useful when you need to solve equations involving exponentials, especially when the exponent is unknown.

• Base: The number we raise to a power to get the other number (here, 64).
• Result: The number we obtain after applying the power to the base (here, 4).
• Logarithm Value: The power or exponent itself (here, \(\frac{1}{3}\)).

By converting exponential equations into logarithmic form, we can solve for unknowns and better understand relationships between numbers.

Mathematical Expression

A mathematical expression is a phrase that can include numbers, variables, and operations to represent a value or relationship. An expression does not have an equality sign like an equation does; it's more about showing a calculation.

For example, \(64^{1/3}\) is a mathematical expression that represents the calculation of the cube root of 64. Mathematical expressions can take many forms, such as sums, differences, products, or powers.

• Components: Numbers, operators, variables, and sometimes constants.
• Purpose: To represent a particular quantity or relationship.
• Versatility: Expressions can be manipulated and transformed into equivalent forms, such as rewriting them as logarithmic equations.

Understanding how to manipulate mathematical expressions allows one to solve complex problems effectively by breaking them down into more manageable forms.