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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 41
Capitolo 5, Problema 41

Evaluate each expression without using a calculator. 8log8198^{\(\log\)_8 19}

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1
Recognize that the expression is of the form \(a^{\log_a b}\), where the base of the exponent and the base of the logarithm are the same (in this case, 8).
Recall the logarithmic identity: \(a^{\log_a b} = b\). This identity holds because the logarithm \(\log_a b\) is the exponent to which you raise \(a\) to get \(b\).
Apply this identity directly to the expression \(8^{\log_8 19}\), which simplifies to just 19.
Understand that this simplification works without any calculation because the logarithm and the exponent base cancel each other out.
Therefore, the value of the expression \(8^{\log_8 19}\) is simply 19.

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Logarithmic and Exponential Functions

Logarithms and exponents are inverse operations. The logarithm log_b(a) answers the question: to what power must the base b be raised to get a? Understanding this inverse relationship is key to simplifying expressions like b^(log_b(x)).
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Graphs of Logarithmic Functions

Properties of Logarithms and Exponents

One important property is that b^(log_b(x)) = x, where b is the base of both the exponent and the logarithm. This property allows simplification of expressions where the base of the exponent matches the base of the logarithm.
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Change of Base Property

Evaluating Expressions Without a Calculator

When evaluating expressions like 8^(log_8 19) without a calculator, use algebraic properties to simplify rather than compute directly. Recognizing patterns and applying properties reduces complex expressions to simpler forms.
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Evaluating Algebraic Expressions
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