Question

Prove the power rule of logarithms.

   Prove the power rule of logarithms.

Precalculus: Graphical, Numerical, Algebraic

Precalculus: Graphical, Numerical, Algebraic

Franklin D.… 8th Edition

Chapter 3, Problem 38

Instant Answer

Solved by Expert Amrita Bhasin

07/12/2019

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Step 1: First, we start with the exponential form of a logarithm, which is $a = c^{x}$. Show more…

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Prove the power rule of logarithms.

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Key Concepts

Definition of Logarithms

A logarithm is the inverse function to exponentiation. It is formally defined so that for a positive base b (with b ? 1) and a positive number x, the logarithm log_b(x) is the exponent to which b must be raised to yield x, i.e., b^(log_b(x)) = x. This fundamental definition is crucial for proving all logarithmic identities.

Exponential Functions

Exponential functions are functions in which the variable appears as the exponent. They have the form b^x, where b is a positive constant. The inverse relationship between logarithms and exponential functions allows us to move freely between the two, providing a framework for manipulating and proving logarithmic properties like the power rule.

Properties of Exponents

The properties of exponents, such as the rule (b^a)^c = b^(a*c), play a central role in the derivation of logarithmic identities. This particular exponent rule allows one to understand how powers combine and is directly used in proving the power rule for logarithms by showing that raising a number to a power and then taking the logarithm is equivalent to multiplying the logarithm by that power.

Power Rule of Logarithms

The power rule of logarithms states that log_b(x^r) = r * log_b(x) for any real number r. The proof involves starting with the definition of the logarithm and applying the properties of exponents. By expressing x^r through its exponential representation and using the exponentiation rule, the multiplicative factor r can be shown to emerge naturally when taking the logarithm, thereby confirming the power rule.

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