Solve the following equations and inequalities. 1. log⁡(3x−1)=log⁡(4−x) 2. log2x3=log2(x) 3. ln⁡8−x2=ln⁡(2−x) 4. log518−x2=log5(6−x)

Expert answers to "Solve the following equations and inequalities. 1. log⁡(3x−1)=log⁡(4−x)..."

High School
Algebra
Pre-Algebra
Equations, expressions, and inequalitie

ddaeeric

Answered question

2021-10-23

Solve the following equations and inequalities.
1.

\log (3 x - 1) = \log (4 - x)

\log_{2} x^{3} = \log_{2} (x)

\ln 8 - x^{2} = \ln (2 - x)

\log_{5} 18 - x^{2} = \log_{5} (6 - x)

Answer & Explanation

oppturf

Skilled2021-10-24Added 94 answers

1) $\log (3 x - 1) = \log (4 - x)$
removing log on sides,
$\Rightarrow 3 x - 1 = 4 - x$
$\Rightarrow 4 x = 6$
$\Rightarrow x = \frac{5}{4}$
2) $\log_{2}^{x^{3}} = \log_{2}^{x}$
$\Rightarrow 3 \log_{2}^{x} = \log_{2}^{x} [∵ \log_{c}^{a^{b}} = b \log_{c}^{a}]$
Which is incorrect so correct eq. is
$\log_{2}^{{(x)}^{3}} = 3 \log_{2}^{x}$
3) $\ln (8 - x^{2}) = \ln (2 - x)$
removing ln on both sides,
$\Rightarrow 8 - x^{2} = 2 - x$
$\Rightarrow x^{2} - x - 6 = 0$
$\Rightarrow (x - 3) (x + 2) = 0$
$\Rightarrow x = 3 or x = - 2$
4) $\log (18 - x^{2}) = \log_{5}^{(6 - x)}$
removing log on both sides,
$\Rightarrow 18 - x^{2} = 6 - x$
$\Rightarrow x^{2} - x - 12 = 0$
$\Rightarrow (x - 4) (x + 3) = 0$
$\Rightarrow x = 4 or x = - 3$

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