Log sqrt x^2+1=1/5
Find x
log(sqrt(x^2+1)) = 1/5
1/2 log(x^2+1) = 1/5
log(x^2+1) = 2/5
x^2+1 = e^(2/5)
...
To find the value of x in the equation log(sqrt(x^2 + 1)) = 1/5, we can follow these steps:
Step 1: Rewrite the equation
First, let's rewrite the equation using the logarithmic property which states that log(a^b) = b * log(a):
1/5 = log(sqrt(x^2 + 1))
Step 2: Convert the logarithmic equation to an exponential equation
To convert the equation from logarithmic to exponential form, we need to exponentiate both sides using the base of the logarithm. In this case, the base is 10:
10^(1/5) = sqrt(x^2 + 1)
Step 3: Simplify the equation
Raising both sides of the equation to the power of 5 gives us:
(10^(1/5))^5 = (sqrt(x^2 + 1))^5
Simplifying further:
10 = (x^2 + 1)^(5/2)
Step 4: Isolate x
To isolate x, we need to undo the square root and the exponentiation on the right side of the equation. We will raise both sides to the power of 2/5:
10^(2/5) = (x^2 + 1)
Step 5: Solve for x
Subtract 1 from both sides of the equation:
10^(2/5) - 1 = x^2
Now, take the square root of both sides to get rid of the square:
sqrt(10^(2/5) - 1) = x
So, the value of x is equal to the square root of (10 raised to the power of 2/5) minus 1:
x = sqrt(10^(2/5) - 1)
You can use a calculator to evaluate this expression and find the numerical value of x.