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)]

Verified step by step guidance

1. Recall the change of base formula for logarithms: \(\log_a(b) = \frac{\log_c(b)}{\log_c(a)}\) for any positive base \(c \neq 1\).
2. Apply the change of base formula to \(\log_7(3)\) using base 3: \(\log_7(3) = \frac{\log_3(3)}{\log_3(7)}\).
3. Since \(\log_3(3) = 1\), simplify the expression to \(\log_7(3) = \frac{1}{\log_3(7)}\).
4. Notice that this means \(\log_3(7) = \frac{1}{\log_7(3)}\), which matches the original equation given.
5. Therefore, the equation \(\log_3(7) = \frac{1}{\log_7(3)}\) is true, based on the properties of logarithms.

Key Concepts

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.