Problem 18 - Algebra A Combined Function

Question

Write each as a logarithmic equation. $$ 5^{3}=125 $$

Step-by-Step Solution

Answer

\(\log_5(125) = 3\)

Step 1: Understand the Exponential Form

The given equation is an exponential equation: \(5^3 = 125\). Here, 5 is the base, 3 is the exponent, and 125 is the result of raising the base to the power of the exponent.

Step 2: Convert to Logarithmic Form

To write the exponential equation \(5^3 = 125\) as a logarithmic equation, use the definition of a logarithm. The equation \(b^x = y\) can be rewritten in logarithmic form as \(\log_b(y) = x\). For this problem, we identify \(b = 5\), \(x = 3\), and \(y = 125\).

Step 3: Write the Logarithmic Equation

Using the values identified and applying the definition of the logarithmic form, the equation \(5^3 = 125\) becomes \(\log_5(125) = 3\). This is the logarithmic representation of the given exponential equation.

Key Concepts

Exponential Form

The exponential form of an equation is a way to express repeated multiplication of a number by itself. In the equation \(5^3 = 125\), the exponential form shows that the base, which is 5, is multiplied by itself 3 times to give the result, 125.

Here’s how it works:

• The *base* is the number 5.
• The *exponent* is 3, indicating how many times the base is multiplied by itself.
• The *result* is the product of the repeated multiplication, which is 125 in this case.

To understand it simply, think of the exponent as a shortcut for writing out the multiplication: \(5 \times 5 \times 5 = 125\). Exponential form is used extensively in mathematics for simplifying expressions and solving equations.

Logarithmic Form

Logarithmic form is another way to represent the relationship in an exponential equation. In contrast to the exponential form \(b^x = y\), the logarithmic form expresses this relationship as \(\log_b(y) = x\).

Using our example \(5^3 = 125\), it transforms into the logarithmic form \(\log_5(125) = 3\). This clearly states that 3 is the exponent to which the base 5 must be raised to produce the number 125.

The conversion between exponential and logarithmic forms is particularly useful for solving equations where the exponent is unknown. Logarithms help in "undoing" exponential functions to solve for the exponent.

Base of Logarithm

The base of a logarithm in the equation \(\log_b(y) = x\) is the number \(b\). It is the foundational value that is raised to the power, indicated by the logarithm, to produce the given number \(y\).

In our example \(\log_5(125) = 3\), the base is 5. This base indicates: "5 raised to which power (the exponent) will result in 125?"

• The base is a critical part of both exponential and logarithmic equations.
• It dictates the "scale" or "different sizes" of results based on the exponent chosen.

Choosing different bases provides different scales for interpreting logarithms, making them versatile tools in handling diverse mathematical contexts.

Definition of Logarithm

The definition of a logarithm provides a formal way to understand the transformation between exponential and logarithmic forms. Simply put, a logarithm answers the question: "To what power must the base be raised, to yield a specific number?"

By definition, if \(b^x = y\), then \(\log_b(y) = x\). So, for the exponential equation \(5^3 = 125\), you have \(\log_5(125) = 3\). The logarithm here, \(\log_5(125)\), determines the number 3 – the power needed to raise 5 to obtain 125.

• Logarithms thus transform multiplicative processes into additive ones, simplifying complex calculations.
• Understanding this transformation is crucial for solving equations where the unknown is an exponent.

Through their unique definitions, logarithms extend their utility beyond simple exponentiation, offering essential insights in various applications such as science, economics, and engineering.