Help me solve this logarithm question

[raanikeri]

I checked the answer and it says

I don't really understand this.
Can anyone explain this to me? I couldn't find any online videos to help me understand this either.

I do know how to convert from index form to logarithmic form and simplify logarithmic expressions, solve some logarithmic equations. I don't get how to sketch the graph to find the answer in these 2 questions though.

 

[e.jane.aran]

View attachment 4128
View attachment 4130

I checked the answer and it says View attachment 4129

View attachment 4131

I don't really understand this.
...I don't get how to sketch the graph to find the answer...

Sketch y_1 = log(x). On the same axis system, sketch y_2 = 5 - x. Find where the two lines intersect.

 

[HallsofIvy]

If you want a more "analytic" (non-"sketch") take the exponential of both sides:
x= e^{5- x}= e^5e^{-x}
Multiply both sides by e^x:
xe^x= e^5.

Apply the "Lambert W" function (defined as the inverse function to xe^x) to both sides:
x= W(e^5).

 

[ROBERTMILLS]

To solve these equations, we'll need to consider different cases for the domain of x and handle each case separately.

For Equation (I): log(x)=5−x

We'll solve this by transforming the equation into a form where we can use numerical methods like graphing or numerical approximation.

1. We can rewrite the equation as log(x)+x−5=0.
2. Let f(x)=log(x)+x−5.
3. We'll find where f(x)=0.

Unfortunately, this equation does not have an algebraic solution, so we'll need to use numerical methods like graphing or iteration to find approximate solutions.

For Equation (II): log(x)=∣5−x∣

We'll solve this by considering two cases: when 5−x is positive and when it's negative.

Case 1: 5−x≥0

In this case, ∣5−x∣=5−x. So the equation becomes log(x)=5−x, which we already solved in Equation (I).

Case 2: 5−x<0

In this case, ∣5−x∣=−(5−x)=x−5. So the equation becomes log(x)=x−5.

Let's solve this equation:

1. We can rewrite the equation as log(x)−x+5=0.
2. Let g(x)=log(x)−x+5.
3. We'll find where g(x)=0.

Again, this equation does not have an algebraic solution, so we'll need to use numerical methods.

In summary, Equation (I) does not have an algebraic solution, and Equation (II) requires solving a non-linear equation using numerical methods due to the presence of the logarithmic function.

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