.5ln(x+3)-lnx=0
Thanks for any help!!! :)
n*ln(a)=ln(a^n)
0.5*ln(x+3)=ln[(x+3)^0.5]=
ln[sqroot(x+3)]
ln(a)-ln(b)=ln(a/b)
5ln(x+3)-ln(x)=ln(sqroot(x+3)/x]=0
ln(1)=0
That mean:
sqroot(x+3)/x=1
sqroot(x+3)=x Square that
x+3=x^2
-x^2+x+3=0
Now you must solve this quadratic equation.
In google type:
quadratic equation online
When you see list of results click on:
Free Online Quadratic Equation Solver.Solve by Quadratic Formula
When page be open in rectacangle type:
-x^2+x+3=0
and click option solve it
A function have two solutions.
ln(negative number)=undefined
So your solution is:
x=(1/2)*[(1+sqroot(13)]
x=2.3027756377319946465596106337352
0.5ln(x+3)-ln(x)=ln(sqroot(x+3)/x]=0
To solve the equation .5ln(x+3) - lnx = 0, we can use the properties of logarithms to simplify the equation and solve for x. Let's go through the steps:
Step 1: Use the properties of logarithms to combine the two logarithms on the left side of the equation into a single logarithm. The properties we will use are:
a) ln(a) - ln(b) = ln(a/b)
b) ln(a^n) = n*ln(a)
Applying property (a), we have:
ln((x+3)/(x)) = 0
Step 2: Since ln(a) = 0 if and only if a = 1, we can set the argument of the logarithm equal to 1 and solve for x:
(x+3)/(x) = 1
Step 3: Multiply both sides of the equation by x to eliminate the denominator:
x + 3 = x
Step 4: Subtract x from both sides of the equation to isolate x:
3 = 0
Step 5: Since 3 does not equal to 0, this equation has no solution.
Therefore, the equation .5ln(x+3) - lnx = 0 has no real solutions.