Question

Prove the properties of logarithms.

   Prove the properties of logarithms.

Schaum's outlines: precalculus

Schaum's outlines: precalculus

Safier, Fred 2nd Edition

Chapter 18, Problem 10

Instant Answer

Solved by Expert Amrita Bhasin

07/12/2019

Step 1

To prove this property, start with the equation log_b(xy) = n. This means that b^n = xy. Rewrite this equation as b^n = b^log_b(x) * b^log_b(y). Since the bases are the same, the exponents can be added, so b^n = b^(log_b(x) + log_b(y)). Therefore, log_b(xy) = Show more…

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Prove the properties of logarithms.

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Key Concepts

Definition of Logarithm

A logarithm is defined as the inverse of exponentiation. Specifically, for a positive base (not equal to 1) and a positive number, the logarithm log?(x) is the exponent to which the base must be raised to yield x. This definition underlies the derivation of all logarithmic properties by directly relating them to the rules of exponents.

Inverse Relationship with Exponentiation

Since logarithms are the inverse function of exponentiation, properties of exponentials (such as a^(m+n) = a^m · a^n, a^(m-n) = a^m / a^n, and (a^m)^k = a^(mk)) can be translated into corresponding logarithmic properties. This inverse nature is fundamental when proving the logarithmic rules.

Product Rule

The product rule for logarithms states that the logarithm of a product is equal to the sum of the logarithms of the individual factors, that is, log?(xy) = log?(x) + log?(y). This property is a direct consequence of the exponent rule that relates multiplication to addition in the exponent, reinforcing the connection between logarithms and exponentiation.

Quotient Rule

The quotient rule establishes that the logarithm of a quotient is the difference of the logarithms of the numerator and the denominator: log?(x/y) = log?(x) - log?(y). This rule is derived from the corresponding exponent rule that details how division translates to subtraction in the power domain.

Power Rule

The power rule for logarithms states that the logarithm of a number raised to a power is equal to the exponent multiplied by the logarithm of the number: log?(x?) = k · log?(x). This property illustrates how exponents can be 'brought down' as coefficients, mirroring the exponential identity (a?)? = a^(mk).

Change of Base Formula

The change of base formula allows the expression of a logarithm in one base in terms of logarithms in another base: log?(x) = log_b(x) / log_b(a), where b is any positive number (? 1). This concept is particularly useful for both theoretical proofs and practical computations, as it enables the analysis and calculation of logarithms across different bases.

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