Problem 40 - College Algebra with Corequisite Support

Question

Can the power property of logarithms be derived from the power property of exponents using the equation \(b^{x}=m ?\) If not, explain why. If so, show the derivation.

Step-by-Step Solution

Answer

Yes, the power property of logarithms can be derived from exponents.

Step 1: Understand the Power Property of Exponents

The power property of exponents states that when you raise a power to another power, you multiply the exponents. In mathematical terms, it can be expressed as \((a^m)^n = a^{m \cdot n}\). This property applies to any real number \(a\), and exponents \(m\) and \(n\).

Step 2: Write the Given Expression

The given equation is \(b^x = m\), where \(b\) is the base, \(x\) is the exponent and \(m\) is the result or power. We need to derive the power property of logarithms from this expression, which requires an understanding of how logarithms and exponents are inversely related.

Step 3: Take Logarithm of Both Sides

To derive the power property of logarithms, take the logarithm of both sides of the equation \(b^x = m\). Applying the logarithm on the left side gives us \(\log(b^x) = \log(m)\).

Step 4: Apply the Logarithm Power Rule

By the power property of logarithms, we know \(\log(a^b) = b \cdot \log(a)\). Applying this property to \(\log(b^x)\), we get \(x \cdot \log(b) = \log(m)\). This shows that the exponent \(x\) can be brought down as a multiplier.

Step 5: Conclusion

From these steps, we see that the relationship between exponents and logarithms allows us to derive the power property of logarithms \(\log(b^x) = x \cdot \log(b)\) using the equation \(b^x = m\). Thus, the power property of logarithms can indeed be derived from the properties of exponents in this context.

Key Concepts

Power Property of Exponents

The power property of exponents is a fundamental concept in mathematics. It allows us to simplify expressions where a power is raised to another power.
In mathematical language, it is expressed as \((a^m)^n = a^{m \cdot n}\).
This property indicates that when we have an exponentiation inside another exponentiation, we simply multiply the exponents.This principle has several benefits:

• Simplifies complex expressions
• Makes calculations quicker and more efficient
• Serves as a building block for more advanced algebraic understanding

As a result, understanding the power property of exponents is crucial when dealing with equations and expressions in mathematics.

Logarithm Power Rule

The logarithm power rule is closely related to the power property of exponents.It offers a way to simplify logarithmic expressions by moving the exponent from the argument to the front of the logarithm as a multiplier. The rule states \(\log(a^b) = b \cdot \log(a)\).
This means if you are logging a power, you can take the exponent and multiply it outside the logarithm.The advantages include:

• Converting complex logarithmic expressions into simpler linear ones
• Making calculations more manageable
• Aiding in solving logarithmic equations

By practicing the application of this rule, students develop a deeper understanding of how logarithms can be manipulated for easy computations.

Inverse Relationship of Exponents and Logarithms

Exponents and logarithms are inverse operations, much like addition and subtraction or multiplication and division.
Understanding this relationship is key in algebra and higher mathematics.If an exponent tells you how many times to multiply a base by itself, a logarithm tells you what the exponent is when a number is expressed as a power of another number.Here’s how the inverse relationship works:

• If \(b^x = m\), then \(\log_b(m) = x\)
• Exponents "undo" logarithms and vice versa
• Using logarithms, one can solve for exponents in equations of the form \(b^x = m\)

This inverse relationship is crucial in making the derivation of properties like the logarithm power rule possible. Grasping this concept allows students to move fluidly between exponential and logarithmic forms, solving equations more efficiently.