Respuesta :
Answer:
From step 4 to 9 is the required steps shown below.
Step-by-step explanation:
Given : Solution below,
[tex]\log x -\log_5 3 = 2\log_5 3\\\log x = 3\log_5 3\\\log x = \log_5 33\\x = 27[/tex]
To find : The error in the solution and what additional step needs to be completed?
Solution : The mistake is the step 3 and additional step need to be added after it.
Step 1 - [tex]\log x -\log_5 3 = 2\log_5 3[/tex]
Step 2 - [tex]\log x = 3\log_5 3[/tex]
Step 3 - [tex]\log x =\log_5 3^3[/tex]
Step 4 - [tex]\log x=\log_5 27[/tex]
Applying change base rule of logarithmic, [tex]\log_b a=\frac{\log a}{\log b}[/tex]
Step 5 - [tex]\log x=\frac{\log 27}{\log 5}[/tex]
Step 6 - [tex]\log x=\frac{1.431}{0.698}[/tex]
Step 7 - [tex]\log x=2.050[/tex]
Step 8 - [tex]x=10^{2.050}[/tex]
Step 9 - [tex]x=112.20[/tex]
From step 4 to 9 is the required steps.
We need to add two steps to the solution
- ln(x) = ln(10)*ln(3^3)/ln(5) = 4.72
- x = Exp(4.72).
Solving the equation step by step.
We start with the equation:
log(x) - log₅(3) = 2*log₅(3).
Where we assume:
log(x) = log₁₀(x) = ln(x)/log(10)
log₅(3) = ln(3)/ln(5)
Now let's follow the steps:
step 1)
log(x) = 3*log₅(3).
Here we just moved the constant term from the left side to the right side, this is correct.
step 2)
log(x) = log₅(3^3).
This is correct, the coefficient can be moved to the exponent of the argument.
step 3)
x = 27
This is wrong, here you say that the arguments must be equal, but the logarithms are different logarithms. so this is incorrect.
Here we should add step 2.5)
ln(x) = ln(10)*ln(3^3)/ln(5)
Now we have:
ln(x) = ln(10)*log(27)/log(5) = 4.72
Now another step, let's say step 2.75)
We apply the exponential equation to both sides:
Exp(ln(x)) = Exp(4.72)
x = 112.17
So we need to add two steps to the solution before step 3.
If you want to learn more about logarithmic equations, you can read:
https://brainly.com/question/10727370
