Problem 259 - Calculus Volume 1

Question

Write the equation in equivalent logarithmic form. \(\sqrt[3]{64}=4\)

Step-by-Step Solution

Answer

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

Step 1: Identify the Exponential Form

The equation given is \( \sqrt[3]{64} = 4 \). This can be interpreted as \( 64^{\frac{1}{3}} = 4 \). This is the exponential form, where we have a base of 64 raised to the power of \( \frac{1}{3} \), resulting in 4.

Step 2: Identify the Base for Logarithmic Form

In logarithmic form, the base of the exponent becomes the base of the logarithm. In \( 64^{\frac{1}{3}} = 4 \), the base is 64. Therefore, the base of the logarithm will also be 64.

Step 3: Write the Logarithmic Form

The logarithmic form of an exponential equation \( a^b = c \) is \( \log_a(c) = b \). Using this, we rewrite \( 64^{\frac{1}{3}} = 4 \) as \( \log_{64}(4) = \frac{1}{3} \).

Key Concepts

Exponential Equations

Exponential equations are mathematical expressions where a constant base is raised to a variable exponent. These equations represent growth or decay patterns observed in various real-world scenarios like population growth or radioactive decay.
In any exponential equation, you typically have a form like \(a^b = c\), where \(a\) is the base, \(b\) is the exponent, and \(c\) is the result. For instance, when you see \(64^{1/3} = 4\), it means 64 raised to the power of one-third equals 4.
To solve exponential equations, it helps to familiarize yourself with exponent rules and properties, such as the fact that \(x^{m/n} = \sqrt[n]{x^m}\). This fundamental understanding aids in converting exponential equations to other forms, like logarithms.

Logarithms

Logarithms are the opposite, or inverse, of exponents. Essentially, they help you find the power to which a base number is raised to get a certain value.
When translating from an exponential equation to a logarithmic one, we use the form \(\log_a(c) = b\). This translates into saying "in what power must the base \(a\) be raised to result in \(c\)?"

• The base of the logarithm corresponds to the base of the exponent in the original equation.
• The argument of the logarithm is the result of the exponentiation, not the exponent itself.

For the equation \(64^{1/3} = 4\), we rewrite it as \(\log_{64}(4) = \frac{1}{3}\). Here, we are expressing the idea that 64 must be raised to the power of one-third to yield 4.

Exponents

Exponents are a way to express repeated multiplication compactly. They consist of a base and a power, indicating how many times to multiply the base by itself.
For example, \(2^3\) means 2 multiplied by itself three times, which equals 8. When you see \(64^{1/3}\), it means you're finding a number that, when cubed, equals 64.
Understanding exponents is crucial, as they are foundational in algebra. Here are some basic rules:

• \(x^m \cdot x^n = x^{m+n}\)
• \((x^m)^n = x^{m \cdot n}\)
• \(x^{-m} = 1/x^m\) indicating an inverse operation.

Remember, fractional exponents like \(1/3\) indicate root operations, where \(64^{1/3}\) stands for the cube root of 64.