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Ch. 4 - Exponential and Logarithmic Functions
Blitzer - College Algebra 8th Edition
Blitzer8th EditionCollege AlgebraISBN: 9780136970514Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoBlitzer 8th EditionCh. 4 - Exponential and Logarithmic FunctionsProblema 33
Capitolo 5, Problema 33

Evaluate each expression without using a calculator. log64 8

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Recognize that the expression \( \log_{64} 8 \) asks for the exponent \( x \) such that \( 64^x = 8 \).
Express both 64 and 8 as powers of the same base. Since both are powers of 2, write \( 64 = 2^6 \) and \( 8 = 2^3 \).
Rewrite the equation \( 64^x = 8 \) as \( (2^6)^x = 2^3 \).
Use the power of a power property to simplify the left side: \( 2^{6x} = 2^3 \).
Since the bases are the same, set the exponents equal: \( 6x = 3 \), then solve for \( x \) by dividing both sides by 6.

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Logarithm Definition

A logarithm answers the question: to what exponent must the base be raised to produce a given number? For example, log_b(a) = c means b^c = a. Understanding this definition is essential to evaluate logarithmic expressions.
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Logarithms Introduction

Change of Base and Prime Factorization

To evaluate logarithms without a calculator, express both the base and the argument as powers of the same prime number. This allows rewriting the logarithm in terms of simpler exponents, facilitating easier calculation.
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Change of Base Property

Properties of Logarithms

Logarithm properties, such as log_b(m^n) = n*log_b(m), help simplify expressions. Applying these rules can break down complex logarithms into manageable parts, making evaluation straightforward.
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Change of Base Property
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