First slide

Trigonometric Identities

Question

If $\log_{3} \{5 + 4 \log_{3} (x - 1)\} = 2,$ then x is equal to

Moderate

A

2
B

4
C

8
D

16

Solution

We must have x-1>0
$\Rightarrow x > 1 - - - - (i)$
$\begin{array}{l} and 5 + 4 \log_{3} (x - 1) > 0 \\ or 4 \log_{3} (x - 1) > - 5 \\ or \log_{3} (x - 1) > - \frac{5}{4} \\ or x - 1 > 3^{- 5 / 4} \\ or x > 1 + 3^{- 5 / 4} - - - - (i i) \end{array}$
From Eqs. (i) and (ii), we get $x > 1 + 3^{- 5 / 4}$
$\begin{array}{ll} ∴ 5 + 4 \log_{3} (x - 1) = 9 \\ \Rightarrow 4 \log_{3} (x - 1) = 4 \\ or \log_{3} (x - 1) = 1 \\ or x - 1 = 3 or x = 4 \end{array}$

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