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.
log_b (xy)⁵ = (log_b x + log_b y)⁵

검증된 단계별 안내

Recall the logarithm property for the logarithm of a product:

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

. This means the log of a product is the sum of the logs.

Apply the power rule of logarithms:

\log b (a)^{n} = n \log b a

. This means the exponent can be brought in front as a multiplier.

Rewrite the left side of the equation using these properties:

\log b (x y)^{5} = 5 \log b (x y)

Next, use the product rule inside the logarithm:

5 \log b (x y) = 5 (\log b x + \log b y)

Compare this with the right side of the original statement:

(\log b x + \log b y)^5

. Notice that the original statement raises the sum to the fifth power, which is different from multiplying the sum by 5. Therefore, the original statement is false, and the correct statement is

\log b (x y)^{5} = 5 (\log b x + \log b y)

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Properties of Logarithms

Logarithms have specific properties that simplify expressions, such as the product rule: log_b(xy) = log_b(x) + log_b(y), and the power rule: log_b(x^n) = n * log_b(x). Understanding these rules is essential to manipulate and evaluate logarithmic expressions correctly.

Exponentiation and Logarithms

When an exponent is applied to a product inside a logarithm, the power rule allows the exponent to be factored out: log_b((xy)^5) = 5 * log_b(xy). This differs from raising the sum of logarithms to a power, which is not equivalent, highlighting the importance of correctly applying exponent rules in logarithmic contexts.

Distinguishing Between Expressions and Their Equivalents

It is crucial to recognize that (log_b x + log_b y)^5 is not the same as 5 * (log_b x + log_b y). The former raises the entire sum to the fifth power, while the latter multiplies the sum by 5. This distinction helps avoid common mistakes when simplifying or verifying logarithmic statements.

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 (xy)5 = (logb x + logb y)5