Let's solve this step by step.

Problem Breakdown:

We are tasked to find an approximation for $\ln\left(\frac{3}{2}\right)$ using the tangent line to the function $f(x) = \ln(x)$ at $x = 1$. This involves using linear approximation (or the tangent line approximation).

Formula for Linear Approximation:

The equation of the tangent line at $x = a$ is: $L(x) = f(a) + f'(a)(x - a)$

Here:

• $f(x) = \ln(x)$
• $a = 1$
• The approximation is for $x = \frac{3}{2}$.

Step 1: Compute $f(a) = f(1)$

$f(1) = \ln(1) = 0$

Step 2: Compute $f'(x)$, the derivative of $f(x) = \ln(x)$

$f'(x) = \frac{1}{x}$ Thus, at $x = 1$: $f'(1) = \frac{1}{1} = 1$

Step 3: Write the Tangent Line Equation

Using the formula for $L(x)$: $L(x) = f(1) + f'(1)(x - 1)$ Substituting the values: $L(x) = 0 + 1(x - 1) = x - 1$

Step 4: Approximate $\ln\left(\frac{3}{2}\right)$

Substitute $x = \frac{3}{2}$ into the tangent line equation $L(x) = x - 1$: $L\left(\frac{3}{2}\right) = \frac{3}{2} - 1 = \frac{1}{2}$

Thus, the approximation for $\ln\left(\frac{3}{2}\right)$ is: $\ln\left(\frac{3}{2}\right) \approx \frac{1}{2}$

Final Answer:

$\boxed{\frac{1}{2}}$

Let me know if you want details or have questions! 😊

Related Questions:

1. How do you derive the linear approximation formula for general functions?
2. What is the exact value of $\ln\left(\frac{3}{2}\right)$, and how does it compare to the approximation?
3. How can the tangent line approximation be used for other values of $x$?
4. What happens to the accuracy of the approximation as $x$ moves farther from the point of tangency?
5. How do you compute derivatives of logarithmic functions with different bases?

Tip:

For functions with rapidly changing slopes, the tangent line approximation works best for values of $x$ very close to the point of tangency!