Pre-Algebra Examples

$f (x) = \ln (x^{2})$ , $[1, e]$

Step 1

Write $f (x) = \ln (x^{2})$ as an equation.

$y = \ln (x^{2})$

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.

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

Step 2.2

Substitute the equation $y = \ln (x^{2})$ for $f (e)$ and $f (1)$ , replacing $x$ in the function with the corresponding $x$ value.

$\frac{(\ln ({(e)}^{2})) - (\ln ({(1)}^{2}))}{(e) - (1)}$

Step 3

Simplify the expression.

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Step 3.1

Simplify the numerator.

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Step 3.1.1

Use logarithm rules to move $2$ out of the exponent.

$\frac{2 \ln (e) - \ln (1^{2})}{e - (1)}$

Step 3.1.2

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

$\frac{2 \cdot 1 - \ln (1^{2})}{e - (1)}$

Step 3.1.3

Multiply $2$ by $1$ .

$\frac{2 - \ln (1^{2})}{e - (1)}$

Step 3.1.4

One to any power is one.

$\frac{2 - \ln (1)}{e - (1)}$

Step 3.1.5

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

$\frac{2 - 0}{e - (1)}$

Step 3.1.6

Multiply $- 1$ by $0$ .

$\frac{2 + 0}{e - (1)}$

Step 3.1.7

Add $2$ and $0$ .

$\frac{2}{e - (1)}$

Step 3.2

Multiply $- 1$ by $1$ .

$\frac{2}{e - 1}$