Question

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. ﻿log⁡6(x−1x2+4)=log⁡6(x−1)−log⁡6(x2+4)\$$log_6$$ \left( \frac{x - 1}{$$x^2 + 4$$} \right) = \$$log_6 (x - 1$$) - \$$log_6$$ $$(x^2 + 4)
log6$$​(x2+4x−1​)=log6​(x−1)−log6​(x2+4)

Accepted Answer

Recall the logarithmic property that states: $$\log_b \left( \frac{M}{N} ight) = \log_b M - \log_b N$$, where $$b \u003e 0$$, $$b 
eq 1$$, and $$M, N \u003e 0. $$Identify the components in the given equation: the left side is $$\log_6$ \left( \frac{x - 1}{x^2$ + 4} ight)$$, and the right side is $$\log_6$ (x - 1) - \log_6$ (x^2$ + 4). $$Check the domain restrictions for the logarithms: since the arguments must be positive, set $$x - 1 \u003e 0$$ and $$x^2 + 4$$ \u003e 0$$. Note that x^2$ + 4 is $$always positive for all real $$x$$, so the main restriction is $$x \u003e 1. $$Since the domain restrictions are satisfied and the logarithmic property applies, conclude that the equation is true for $$x \u003e 1. If $$the problem required a change and the equation was false, you would adjust the right side to match the logarithmic property, but in this case, the equation is already correctly stated.

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. $\(\log\)_6 \(\left\)( \(\frac{x - 1}{x^2 + 4}\) \(\right\)) = \(\log\)_6 (x - 1) - \(\log\)_6 (x^2 + 4)$

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Key Concepts

Properties of Logarithms

Domain of Logarithmic Functions

Equation Verification and Transformation