Ch. 4 - Exponential and Logarithmic Functions

Blitzer8th EditionCollege AlgebraISBN: 9780136970514당신이 사용하는 게 아니라요?교과서 변경

Blitzer 8th Edition

Ch. 4 - Exponential and Logarithmic Functions

문제 127

5장, 문제 127

In Exercises 125–128, determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement.
$l o g_{b} (x^{3} + y^{3}) = 3 l o g_{b} x + 3 l o g_{b} y$

검증된 단계별 안내

Recall the logarithm property for products:

\log b (x \(\cdot\) y) = \log b (x) + \log b (y)

. This means logarithms turn multiplication inside the log into addition outside the log.

Notice that the expression inside the logarithm on the left side is

x^{3} + y^{3}

, which is a sum, not a product. The logarithm property for sums is not the same as for products.

The right side of the equation is

3 \(\log\)_b x + 3 \(\log\)_b y

, which can be rewritten using the power rule of logarithms as

\(\log\)_b (x^3) + \(\log\)_b (y^3)

. Using the product rule, this equals

\(\log\)_b (x^3 y^3)

Since

\(\log\)_b (x^3 + y^3)

is not equal to

\(\log\)_b (x^3 y^3)

, the original statement is false.

To make the statement true, replace the sum inside the logarithm with a product:

\(\log\)_b (x^3 y^3) = 3 \(\log\)_b x + 3 \(\log\)_b y

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주요 개념

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Properties of Logarithms

Logarithms have specific properties that simplify expressions, such as the product rule: log_b(MN) = log_b M + log_b N, and the power rule: log_b(M^k) = k log_b M. Understanding these rules is essential to manipulate and evaluate logarithmic expressions correctly.

Sum of Cubes vs. Product

The expression x^3 + y^3 is a sum of cubes, which cannot be factored into a simple product of x and y terms. Since logarithm properties apply to products, not sums, recognizing the difference between sums and products is crucial to avoid incorrect application of log rules.

True or False Statements in Algebra

Determining the truth of algebraic statements involves verifying if the given equality holds for all valid values. If false, one must identify the error and correct it, often by applying the correct algebraic or logarithmic rules, ensuring the statement accurately reflects mathematical principles.

In Exercises 125–128, determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement.logb(x3+y3)=3logbx+3logbylog_{b}(x^3+y^3)=3log_{b}x+3log_{b}y