what is the value of log subscript 5 baseline 125

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14-02-2025

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The expression log₅ 125 represents a logarithm, specifically asking: "To what power must we raise the base 5 to obtain 125?" Understanding logarithms is crucial to solving this and similar problems. Let's break it down.

Understanding Logarithms

A logarithm is the inverse operation of exponentiation. In the general form logₐ b = x, we have:

• a: the base (the number being raised to a power)
• b: the argument (the result of the exponentiation)
• x: the exponent (the power to which the base is raised)

The equation logₐ b = x is equivalent to the exponential equation aˣ = b.

Solving for log₅ 125

In our case, we have log₅ 125. This means:

• a = 5 (the base)
• b = 125 (the argument)
• x = ? (the exponent we need to find)

We can rewrite this logarithmic equation as an exponential equation:

5ˣ = 125

Now, we need to determine what exponent (x) will make this equation true. Since 125 is a power of 5 (125 = 5 x 5 x 5 = 5³), we can see that:

5³ = 125

Therefore, x = 3.

The Answer

The value of log₅ 125 is 3.

Further Exploration: Properties of Logarithms

Understanding the properties of logarithms can simplify more complex calculations. Here are a few key properties:

• Product Rule: logₐ (xy) = logₐ x + logₐ y
• Quotient Rule: logₐ (x/y) = logₐ x - logₐ y
• Power Rule: logₐ xⁿ = n logₐ x
• Change of Base Formula: logₐ x = (logₓ x) / (logₓ a) (This allows you to change the base of a logarithm)

These properties are invaluable for solving more intricate logarithmic equations and expressions. Mastering them will enhance your understanding of exponential and logarithmic functions.

Practical Applications of Logarithms

Logarithms have wide-ranging applications in various fields, including:

• Science: Measuring the intensity of earthquakes (Richter scale), sound (decibels), and acidity (pH scale).
• Engineering: Analyzing signal processing and electrical circuits.
• Finance: Calculating compound interest and exponential growth.
• Computer Science: Analyzing algorithms and data structures.

This article provides a foundational understanding of logarithms and demonstrates how to solve for the value of log₅ 125. Remember to always consider the base and argument carefully when working with logarithms.