what is x and cosh ln(x) when tanh(lnx^(1/2)= 12/13
Do you mean ln√x or √lnx ?
To find the values of x and cosh(ln(x)) when tanh(ln(x)^(1/2)) = 12/13, we can follow these steps:
Step 1: Rewrite the equation:
tanh(ln(x)^(1/2)) = 12/13
Step 2: Square both sides of the equation:
[tanh(ln(x)^(1/2))]^2 = (12/13)^2
Step 3: Simplify the equation:
tanh^2(ln(x)^(1/2)) = 144/169
Step 4: Replace the tanh^2 function with its definition:
[e^(2x) - 1] / [e^(2x) + 1] = 144/169
Step 5: Cross-multiply:
169(e^(2x) - 1) = 144(e^(2x) + 1)
Step 6: Expand the equation:
169e^(2x) - 169 = 144e^(2x) + 144
Step 7: Rearrange the equation:
25e^(2x) = 313
Step 8: Solve for e^(2x):
e^(2x) = 313/25
Step 9: Take the natural logarithm of both sides:
2x = ln(313/25)
Step 10: Solve for x:
x = ln(313/25) / 2
Now, to find the value of cosh(ln(x)), we substitute the value we found for x back into the equation:
cosh(ln(x)) = cosh(ln(ln(313/25) / 2))
At this point, we can use a scientific calculator or a programming language with built-in functions to calculate the value of cosh(ln(x)) or approximately evaluate it.