Solve $\textrm{ln}(4\cdot 2^{3x})+3\cdot \textrm{ln}(16) = \textrm{ln}(512\cdot 2^{x^2+3x+1})$.
Antwoord 1 correct
Correct
Antwoord 2 optie
$x=2$
Antwoord 2 correct
Fout
Antwoord 3 optie
There is no solution.
Antwoord 3 correct
Fout
Antwoord 4 optie
$x=\sqrt{4088}$
Antwoord 4 correct
Fout
Antwoord 1 optie
None of the other answers is correct.
Antwoord 1 feedback
Correct: $$\begin{align*}
\textrm{ln}(4\cdot 2^{3x})+3\cdot \textrm{ln}(16) = \textrm{ln}(512\cdot 2^{x^2+3x+1}) & \Leftrightarrow \textrm{ln}(4\cdot 2^{3x})+ \textrm{ln}(16^{3}) = \textrm{ln}(512\cdot 2^{x^2+3x+1})\\
& \Leftrightarrow \textrm{ln}(16^3\cdot 4\cdot 2^{3x}) = \textrm{ln}(512\cdot 2^{x^2+3x+1})\\
& \Leftrightarrow 16^3\cdot 4\cdot 2^{3x} = 512\cdot 2^{x^2+3x+1}\\
& \Leftrightarrow 2^{12}\cdot 2^2\cdot 2^{3x} = 2^9\cdot 2^{x^2+3x+1}\\
& \Leftrightarrow 2^{3x+14} = 2^{x^2+3x+10}\\
& \Leftrightarrow 3x+14 = x^2+3x+10\\
& \Leftrightarrow x^2-4 = 0\\
& \Leftrightarrow x=-2 \mbox{ or } x=2.
\end{align*}$$
\textrm{ln}(4\cdot 2^{3x})+3\cdot \textrm{ln}(16) = \textrm{ln}(512\cdot 2^{x^2+3x+1}) & \Leftrightarrow \textrm{ln}(4\cdot 2^{3x})+ \textrm{ln}(16^{3}) = \textrm{ln}(512\cdot 2^{x^2+3x+1})\\
& \Leftrightarrow \textrm{ln}(16^3\cdot 4\cdot 2^{3x}) = \textrm{ln}(512\cdot 2^{x^2+3x+1})\\
& \Leftrightarrow 16^3\cdot 4\cdot 2^{3x} = 512\cdot 2^{x^2+3x+1}\\
& \Leftrightarrow 2^{12}\cdot 2^2\cdot 2^{3x} = 2^9\cdot 2^{x^2+3x+1}\\
& \Leftrightarrow 2^{3x+14} = 2^{x^2+3x+10}\\
& \Leftrightarrow 3x+14 = x^2+3x+10\\
& \Leftrightarrow x^2-4 = 0\\
& \Leftrightarrow x=-2 \mbox{ or } x=2.
\end{align*}$$
Antwoord 2 feedback
Wrong: What are the solutions to $x^2=4$?
Try again.
Try again.
Antwoord 3 feedback
Wrong: First of all apply the Properties of logarithmic functions to get on both sides of the equation one ln-term.
See Properties logarithmic functions.
See Properties logarithmic functions.
Antwoord 4 feedback
Wrong: First of all apply the Properties of logarithmic functions to get on both sides of the equation one ln-term.
See Properties logarithmic functions.
See Properties logarithmic functions.