Finite Math Examples

$2 \log (x) + 1 = 5$

Step 1

Write $\log (x^{2}) - 4 = 0$ as a function.

$f (x) = \log (x^{2}) - 4$

Step 2

Subtract $5$ from both sides of the equation.

$2 \log (x) + 1 - 5 = 0$

Step 3

Simplify $2 \log (x) + 1 - 5$ .

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Step 3.1

Simplify $2 \log (x)$ by moving $2$ inside the logarithm.

$\log (x^{2}) + 1 - 5 = 0$

Step 3.2

Subtract $5$ from $1$ .

$\log (x^{2}) - 4 = 0$

Step 4

Consider the difference quotient formula.

$\frac{f (x + h) - f (x)}{h}$

Step 5

Find the components of the definition.

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Step 5.1

Evaluate the function at $x = x + h$ .

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Step 5.1.1

Replace the variable $x$ with $x + h$ in the expression.

$f (x + h) = \log ({(x + h)}^{2}) - 4$

Step 5.1.2

The final answer is $\log ({(x + h)}^{2}) - 4$ .

$\log ({(x + h)}^{2}) - 4$

Step 5.2

Find the components of the definition.

$f (x + h) = \log ({(x + h)}^{2}) - 4$

$f (x) = \log (x^{2}) - 4$

$f (x + h) = \log ({(x + h)}^{2}) - 4$

$f (x) = \log (x^{2}) - 4$

Step 6

Plug in the components.

$\frac{f (x + h) - f (x)}{h} = \frac{\log ({(x + h)}^{2}) - 4 - (\log (x^{2}) - 4)}{h}$

Step 7

Simplify the numerator.

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Step 7.1

Apply the distributive property.

$\frac{\log ({(x + h)}^{2}) - 4 - \log (x^{2}) - - 4}{h}$

Step 7.2

Multiply $- 1$ by $- 4$ .

$\frac{\log ({(x + h)}^{2}) - 4 - \log (x^{2}) + 4}{h}$

Step 7.3

Use the quotient property of logarithms, $\log_{b} (x) - \log_{b} (y) = \log_{b} (\frac{x}{y})$ .

$\frac{\log (\frac{{(x + h)}^{2}}{x^{2}}) - 4 + 4}{h}$

Step 7.4

Add $- 4$ and $4$ .

$\frac{\log (\frac{{(x + h)}^{2}}{x^{2}}) + 0}{h}$

Step 7.5

Add $\log (\frac{{(x + h)}^{2}}{x^{2}})$ and $0$ .

$\frac{\log (\frac{{(x + h)}^{2}}{x^{2}})}{h}$

Step 8