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Solve the logarithmic equation.
In (x+10)-ln x=3
Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
A. x=_________
(Type an integer or decimal rounded to four decimal places as needed. Use a comma to separate answers as needed.)

Sagot :

Answer:

x = 0.5240

Step-by-step explanation:

(Natural) Logarithm Properties

Quotient Property

[tex]\rm log_b(a)-log_b(c)=log_b(\dfrac{a}{c} )[/tex],

where b is the base and a and c are distinct arguments (the number that b raised to a certain power equals to).

Exponential Property (Natural)

[tex]\rm e^l^n^(^x^)=x[/tex],

where e is Euler's Number or the natural constant and ln is the natural logarithm with base of e.

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Solving the Problem

The left hand side follows the quotient property so it can be rewritten into one big natural log.

[tex]\rm ln(\dfrac{x+10}{x} )=3[/tex]

We can't solve for x until the x terms are isolated so, we can use the exponential property to remove the natural log.

                                              [tex]\rm e^l^n^(^\frac{x+10}{x} ^)=e^3[/tex]

                                                 [tex]\rm \dfrac{x+10}{x} =e^3[/tex]

All there's left is to solve for x using algebra!

                                            [tex]\rm x+10=xe^3[/tex]

                                            [tex]\rm 10=xe^3-x[/tex]

                                            [tex]\rm 10=x(e^3-1)[/tex]

                                              [tex]\rm \dfrac{10}{e^3-1} =x[/tex]

                                              [tex]\rm 0.5240=x[/tex]

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