Question

154. The linearization of log₃xa. Find the linearization off(x) = log₃x at x = 3.Then round its coefficients to two decimal places.

Accepted Answer

Identify the function to linearize: $$ f(x) = \log_3 x . $$The goal is to find the linear approximation of $$ f(x) $$ near $$ x = 3 . $$Recall the formula for linearization (or the linear approximation) of a function $$ f(x)  at$$ $$ x = a $$:

$$ L(x) = f(a) + f'(a)(x - a) $$

where $$ f'(a)  is $$the derivative of $$ f(x) $$ evaluated at $$ x = a . $$Calculate $$ f(3) $$: Since $$ f(x) = \log_3 x $$, use the change of base formula to express it in terms of natural logarithms:

$$ f(3) = \log_3 3 = 1 $$

because $$ \log_b b = 1 $$ for any base $$ b \u003e 0 . $$Find the derivative $$ f'(x)  of$$ $$ f(x) = \log_3 x . $$Using the change of base formula:

$$ f(x) = \frac{\ln x}{\ln 3}

so$$

$$ f'(x) = \frac{1}{x \ln 3} .

$$Evaluate this at $$ x = 3 $$:

$$ f'(3) = \frac{1}{3 \ln 3} . $$Write the linearization formula substituting $$ a = 3 $$, $$ f(3) = 1 $$, and $$ f'(3) = \frac{1}{3 \ln 3} $$:

$$ L(x) = 1 + \frac{1}{3 \ln 3} (x - 3) .

$$Finally, round the coefficients $$ 1 $$ and $$ \frac{1}{3 \ln 3}  to $$two decimal places to complete the linearization.

154. The linearization of log₃x
a. Find the linearization of
f(x) = log₃xatx = 3.
Then round its coefficients to two decimal places.

Verified video answer for a similar problem:

Key Concepts

Linearization of a Function

Derivative of Logarithmic Functions

Change of Base and Evaluation at a Point

154. The linearization of log₃xa. Find the linearization off(x) = log₃xatx = 3.Then round its coefficients to two decimal places.