Solve for X. Give exact answer in form of fraction.
ln 27+ln(x-8)=ln(9x-8)
ln 27(x-8) = ln(9x-8)
logs are equal, so values are equal:
27(x-8) = 9x-8
27x - 216 = 9x - 8
18x = 208
x = 104/9
To solve for x in the equation ln 27 + ln(x-8) = ln(9x-8), we can use the properties of logarithms.
Step 1: Simplify the equation using the logarithmic property that ln(a) + ln(b) = ln(a*b).
Using this property, we can rewrite the equation as ln(27 * (x-8)) = ln(9x-8).
Step 2: Rewrite the equation in exponential form.
The logarithmic equation ln(27 * (x-8)) = ln(9x-8) can be rewritten as e^ln(27 * (x-8)) = e^ln(9x-8).
Using the property e^ln(x) = x, we get 27 * (x-8) = 9x-8.
Step 3: Solve the equation for x.
Expanding the equation, we get 27x - 216 = 9x - 8.
Subtracting 9x from both sides gives 18x - 216 = -8.
Adding 216 to both sides gives 18x = 208.
Finally, dividing both sides by 18 gives x = 208/18.
Simplifying the fraction 208/18, we get x = 104/9.
Therefore, the exact value of x in the equation ln 27 + ln(x-8) = ln(9x-8) is x = 104/9 as a fraction.