# How do you use the tangent line approximation to approximate the value of #ln(1003)# ?

##### 1 Answer

The answer is

Another term for tangent line approximation is linear approximation. The linear approximation function is:

#L(x)~~f(a)+f'(a)(x-a)#

So we need to find the derivative:

#f(x)=ln(x)#

#f'(x)=1/x#

Now, we need to pick an

#f(a)=f(10^3)=ln(10^3)=3ln(10)#

#f'(a)=f'(1000)=1/(1000)#

So our linear approximation is:

#L(x)~~3ln(10)+1/(1000)(x-1000)#

#L(1003)~~3ln(10)+1/(1000)(1003-1000)#

#~~3ln(10)+3/(1000)#

#~~3ln(10)+.003#

We should leave this as the answer since it's supposed to be mental math. But let's look at how accurate this is: