Finite Math Examples

$\ln (8 x - 6) = 3$

Step 1

To solve for $x$ , rewrite the equation using properties of logarithms.

$e^{\ln (8 x - 6)} = e^{3}$

Step 2

Rewrite $\ln (8 x - 6) = 3$ in exponential form using the definition of a logarithm. If $x$ and $b$ are positive real numbers and $b \neq 1$ , then $\log_{b} (x) = y$ is equivalent to $b^{y} = x$ .

$e^{3} = 8 x - 6$

Step 3

Solve for $x$ .

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Step 3.1

Rewrite the equation as $8 x - 6 = e^{3}$ .

$8 x - 6 = e^{3}$

Step 3.2

Add $6$ to both sides of the equation.

$8 x = e^{3} + 6$

Step 3.3

Divide each term in $8 x = e^{3} + 6$ by $8$ and simplify.

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Step 3.3.1

Divide each term in $8 x = e^{3} + 6$ by $8$ .

$\frac{8 x}{8} = \frac{e^{3}}{8} + \frac{6}{8}$

Step 3.3.2

Simplify the left side.

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Step 3.3.2.1

Cancel the common factor of $8$ .

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Step 3.3.2.1.1

Cancel the common factor.

$\frac{8 x}{8} = \frac{e^{3}}{8} + \frac{6}{8}$

Step 3.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{e^{3}}{8} + \frac{6}{8}$

Step 3.3.3

Simplify the right side.

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Step 3.3.3.1

Cancel the common factor of $6$ and $8$ .

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Step 3.3.3.1.1

Factor $2$ out of $6$ .

$x = \frac{e^{3}}{8} + \frac{2 (3)}{8}$

Step 3.3.3.1.2

Cancel the common factors.

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Step 3.3.3.1.2.1

Factor $2$ out of $8$ .

$x = \frac{e^{3}}{8} + \frac{2 \cdot 3}{2 \cdot 4}$

Step 3.3.3.1.2.2

Cancel the common factor.

$x = \frac{e^{3}}{8} + \frac{2 \cdot 3}{2 \cdot 4}$

Step 3.3.3.1.2.3

Rewrite the expression.

$x = \frac{e^{3}}{8} + \frac{3}{4}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$x = \frac{e^{3}}{8} + \frac{3}{4}$

Decimal Form:

$x = 3.26069211 \dots$