elsa tries to solve the following equation, and determines there is no solution. is she correct? explain…

Explanation:

Step1: Use log - property to rewrite the equation

First, use the property (y=\log_{a}b\Rightarrow b = a^{y}). Rewrite the given equation (\log_{2}x=\log_{2}(3x + 5)+4) as (\log_{2}x-\log_{2}(3x + 5)=4). Then, by the quotient - rule of logarithms (\log_{a}M-\log_{a}N=\log_{a}\frac{M}{N}), we get (\log_{2}\frac{x}{3x + 5}=4).

Step2: Convert the logarithmic equation to an exponential equation

Converting (\log_{2}\frac{x}{3x + 5}=4) to exponential form using (y=\log_{a}b\Rightarrow b=a^{y}), we have (\frac{x}{3x + 5}=2^{4}=16).

Step3: Solve the resulting rational equation

Cross - multiply the equation (\frac{x}{3x + 5}=16) to get (x = 16(3x+5)). Expand the right - hand side: (x=48x + 80). Subtract (48x) from both sides: (x-48x=80), so (- 47x=80), and (x=-\frac{80}{47}).

Step4: Check the domain of the original logarithmic equation

For the original equation (\log_{2}x) and (\log_{2}(3x + 5)), the arguments of the logarithms must be positive. For (y = \log_{2}x), (x>0), and for (y=\log_{2}(3x + 5)), (3x+5>0) (i.e., (x>-\frac{5}{3})). The solution (x =-\frac{80}{47}\approx - 1.70) does not satisfy the condition (x>0). So, there is no solution.

Answer:

Yes, she is correct.