Algebra Examples

$\ln (5 x + 7) = 8$

Step 1

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

$e^{\ln (5 x + 7)} = e^{8}$

Step 2

Rewrite $\ln (5 x + 7) = 8$ 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 $b^{y} = x$ .

$e^{8} = 5 x + 7$

Step 3

Solve for $x$ .

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Step 3.1

Rewrite the equation as $5 x + 7 = e^{8}$ .

$5 x + 7 = e^{8}$

Step 3.2

Subtract $7$ from both sides of the equation.

$5 x = e^{8} - 7$

Step 3.3

Divide each term in $5 x = e^{8} - 7$ by $5$ and simplify.

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Step 3.3.1

Divide each term in $5 x = e^{8} - 7$ by $5$ .

$\frac{5 x}{5} = \frac{e^{8}}{5} + \frac{- 7}{5}$

Step 3.3.2

Simplify the left side.

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Step 3.3.2.1

Cancel the common factor of $5$ .

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Step 3.3.2.1.1

Cancel the common factor.

$\frac{5 x}{5} = \frac{e^{8}}{5} + \frac{- 7}{5}$

Step 3.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{e^{8}}{5} + \frac{- 7}{5}$

Step 3.3.3

Simplify the right side.

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Step 3.3.3.1

Move the negative in front of the fraction.

$x = \frac{e^{8}}{5} - \frac{7}{5}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$x = \frac{e^{8}}{5} - \frac{7}{5}$

Decimal Form:

$x = 594.79159740 \dots$