Precalculus Examples

$w (x) = \ln (x)$ ; , $1 \leq x \leq 7$

Step 1

Write $w (x) = \ln (x)$ as an equation.

$y = \ln (x)$

Step 2

Substitute using the average rate of change formula.

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Step 2.1

The average rate of change of a function can be found by calculating the change in $y$ values of the two points divided by the change in $x$ values of the two points.

$\frac{f (7) - f (1)}{(7) - (1)}$

Step 2.2

Substitute the equation $y = \ln (x)$ for $f (7)$ and $f (1)$ , replacing $x$ in the function with the corresponding $x$ value.

$\frac{(\ln (7)) - (\ln (1))}{(7) - (1)}$

Step 3

Simplify the expression.

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Step 3.1

Simplify the numerator.

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Step 3.1.1

The natural logarithm of $1$ is $0$ .

$\frac{\ln (7) - 0}{7 - (1)}$

Step 3.1.2

Multiply $- 1$ by $0$ .

$\frac{\ln (7) + 0}{7 - (1)}$

Step 3.1.3

Add $\ln (7)$ and $0$ .

$\frac{\ln (7)}{7 - (1)}$

Step 3.2

Simplify the denominator.

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Step 3.2.1

Multiply $- 1$ by $1$ .

$\frac{\ln (7)}{7 - 1}$

Step 3.2.2

Subtract $1$ from $7$ .

$\frac{\ln (7)}{6}$

Step 3.3

Rewrite $\frac{\ln (7)}{6}$ as $\frac{1}{6} \ln (7)$ .

$\frac{1}{6} \ln (7)$

Step 3.4

Simplify $\frac{1}{6} \ln (7)$ by moving $\frac{1}{6}$ inside the logarithm.

$\ln (7^{\frac{1}{6}})$