Assuming all variables are positive, use properties of logarithms to write the expression as a sum or difference of logarithms or multiples of logarithms.
In(x-3)- In(x-5)= In 5

To solve the equation [tex]\ln(x - 3) - \ln(x - 5) = \ln 5[/tex] using properties of logarithms, we can utilize the quotient rule of logarithms, which states:

[tex] \ln \left(\dfrac{a}{b}\right) = \ln a - \ln b [/tex]

Given:

[tex] \ln(x - 3) - \ln(x - 5) = \ln 5 [/tex]

We can combine the logarithms on the left-hand side using the quotient rule:

[tex] \ln \left(\dfrac{x - 3}{x - 5}\right) = \ln 5 [/tex]

Now, according to the properties of logarithms, if [tex]\ln u = \ln v[/tex], then [tex]u = v[/tex]. Thus, we have:

[tex] \dfrac{x - 3}{x - 5} = 5 [/tex]

Now, solve this equation for [tex]x[/tex]:

[tex] x - 3 = 5(x - 5) [/tex]

[tex] x - 3 = 5x - 25 [/tex]

Now, solve for [tex]x[/tex]:

[tex] 25 - 3 = 5x - x [/tex]

[tex] 22 = 4x [/tex]

[tex] x = \dfrac{22}{4} [/tex]

[tex] x = \dfrac{11}{2} [/tex]

So, the solution to the equation is:

[tex]x = \dfrac{11}{2}[/tex]

Assuming all variables are positive, use properties of logarithms to write the expression as a sum or difference of logarithms or multiples of logarithms. In(x-3)- In(x-5)= In 5

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Assuming all variables are positive, use properties of logarithms to write the expression as a sum or difference of logarithms or multiples of logarithms.
In(x-3)- In(x-5)= In 5