Determine whether each equation is true or false. Where possible, show work to support your conclusion. If the statement is false, make the necessary change(s) to produce a true statement. log₃ (7) = 1/[log₇ (3)]

검증된 단계별 안내

Recall the change of base formula for logarithms: \(\log_a(b) = \frac{\log_c(b)}{\log_c(a)}\) for any positive base \(c \neq 1\).

Apply the change of base formula to \(\log_7(3)\) using base 3: \(\log_7(3) = \frac{\log_3(3)}{\log_3(7)}\).

Since \(\log_3(3) = 1\), simplify the expression to \(\log_7(3) = \frac{1}{\log_3(7)}\).

Notice that this means \(\log_3(7) = \frac{1}{\log_7(3)}\), which matches the original equation given.

Therefore, the equation \(\log_3(7) = \frac{1}{\log_7(3)}\) is true, based on the properties of logarithms.

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

도움이 되었나요?

주요 개념

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

Logarithmic Functions and Notation

A logarithm log_b(a) answers the question: to what power must the base b be raised to get a? Understanding this notation is essential for interpreting and manipulating logarithmic equations.

Change of Base Formula

The change of base formula states that log_b(a) = 1 / log_a(b). This property allows rewriting logarithms with different bases and is key to verifying or transforming logarithmic equations.

Verifying Logarithmic Equations

To determine if a logarithmic equation is true, substitute values or apply logarithmic properties like the change of base formula. Showing work involves rewriting expressions and simplifying to confirm equality.

Determine whether each equation is true or false. Where possible, show work to support your conclusion. If the statement is false, make the necessary change(s) to produce a true statement. log3 (7) = 1/[log7 (3)]