Solve for x 5 natural log of 4x-6=-6

$5 \ln (4 x - 6) = - 6$

Step 1

Divide each term in $5 \ln (4 x - 6) = - 6$ by $5$ and simplify.

Divide each term in $5 \ln (4 x - 6) = - 6$ by $5$ .

$\frac{5 \ln (4 x - 6)}{5} = \frac{- 6}{5}$

Simplify the left side.

Cancel the common factor of $5$ .

Cancel the common factor.

$\frac{5 \ln (4 x - 6)}{5} = \frac{- 6}{5}$

Divide $\ln (4 x - 6)$ by $1$ .

$\ln (4 x - 6) = \frac{- 6}{5}$

Simplify the right side.

Move the negative in front of the fraction.

$\ln (4 x - 6) = - \frac{6}{5}$

Step 2

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

$e^{\ln (4 x - 6)} = e^{- \frac{6}{5}}$

Step 3

Rewrite $\ln (4 x - 6) = - \frac{6}{5}$ 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 $b^{y} = x$ .

$e^{- \frac{6}{5}} = 4 x - 6$

Step 4

Solve for $x$ .

Rewrite the equation as $4 x - 6 = e^{- \frac{6}{5}}$ .

$4 x - 6 = e^{- \frac{6}{5}}$

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$4 x - 6 = \frac{1}{e^{\frac{6}{5}}}$

Add $6$ to both sides of the equation.

$4 x = \frac{1}{e^{\frac{6}{5}}} + 6$

Divide each term in $4 x = \frac{1}{e^{\frac{6}{5}}} + 6$ by $4$ and simplify.

Divide each term in $4 x = \frac{1}{e^{\frac{6}{5}}} + 6$ by $4$ .

$\frac{4 x}{4} = \frac{\frac{1}{e^{\frac{6}{5}}}}{4} + \frac{6}{4}$

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

$\frac{4 x}{4} = \frac{\frac{1}{e^{\frac{6}{5}}}}{4} + \frac{6}{4}$

Divide $x$ by $1$ .

$x = \frac{\frac{1}{e^{\frac{6}{5}}}}{4} + \frac{6}{4}$

Simplify the right side.

Simplify each term.

Multiply the numerator by the reciprocal of the denominator.

$x = \frac{1}{e^{\frac{6}{5}}} \cdot \frac{1}{4} + \frac{6}{4}$

Combine.

$x = \frac{1 \cdot 1}{e^{\frac{6}{5}} \cdot 4} + \frac{6}{4}$

Multiply $1$ by $1$ .

$x = \frac{1}{e^{\frac{6}{5}} \cdot 4} + \frac{6}{4}$

Move $4$ to the left of $e^{\frac{6}{5}}$ .

$x = \frac{1}{4 \cdot e^{\frac{6}{5}}} + \frac{6}{4}$

Cancel the common factor of $6$ and $4$ .

Factor $2$ out of $6$ .

$x = \frac{1}{4 e^{\frac{6}{5}}} + \frac{2 (3)}{4}$

Cancel the common factors.

Factor $2$ out of $4$ .

$x = \frac{1}{4 e^{\frac{6}{5}}} + \frac{2 \cdot 3}{2 \cdot 2}$

Cancel the common factor.

$x = \frac{1}{4 e^{\frac{6}{5}}} + \frac{2 \cdot 3}{2 \cdot 2}$

Rewrite the expression.

$x = \frac{1}{4 e^{\frac{6}{5}}} + \frac{3}{2}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = \frac{1}{4 e^{\frac{6}{5}}} + \frac{3}{2}$

Decimal Form:

$x = 1.57529855 \dots$

Solve for x 5 natural log of 4x-6=-6

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