which logarithmic equation is equivalent to 82 64

Daniel Parker

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23-02-2025

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Understanding the relationship between exponential and logarithmic equations is crucial in mathematics. This article will explore how to express the exponential equation 8² = 64 as an equivalent logarithmic equation. We'll break down the process step-by-step, clarifying the concepts involved.

Understanding Exponential and Logarithmic Equations

An exponential equation shows a variable in the exponent. A simple example is b^x = y. This means "b raised to the power of x equals y."

A logarithmic equation is the inverse of an exponential equation. It essentially asks, "To what power must we raise the base (b) to get y?" This is expressed as log_by = x.

Converting 8² = 64 to a Logarithmic Equation

Let's apply this understanding to the equation 8² = 64. We can identify the base (b), the exponent (x), and the result (y):

• Base (b): 8
• Exponent (x): 2
• Result (y): 64

Using the logarithmic form log_by = x, we can substitute these values:

log₈64 = 2

Therefore, the equivalent logarithmic equation is log₈64 = 2. This reads as "the logarithm of 64 to the base 8 is 2." It means that 8 raised to the power of 2 equals 64.

Visualizing the Relationship

It can be helpful to visualize the relationship between the exponential and logarithmic forms:

Common Logarithms and Natural Logarithms

While the example above uses a specific base (8), there are two common bases used frequently:

• Common Logarithm (base 10): Often written as log(x), this implies a base of 10. For example, log(100) = 2 because 10² = 100.

• Natural Logarithm (base e): Often written as ln(x), where e is Euler's number (approximately 2.718). For example, ln(e) = 1 because e¹ = e.

These are less relevant to directly solving this specific problem, but they represent important concepts in broader logarithmic applications.

Conclusion: The Equivalent Logarithmic Equation

To reiterate, the logarithmic equation equivalent to 8² = 64 is log₈64 = 2. Understanding this conversion is key to mastering exponential and logarithmic functions and their many applications in various fields, such as science, engineering, and finance.