Solve for x 6^(2x+2)*6^(3x)=1

$6^{2 x + 2} \cdot 6^{3 x} = 1$

Step 1

Take the natural logarithm of both sides of the equation to remove the variable from the exponent.

$\ln (6^{2 x + 2} \cdot 6^{3 x}) = \ln (1)$

Step 2

Expand the left side.

Rewrite $\ln (6^{2 x + 2} \cdot 6^{3 x})$ as $\ln (6^{2 x + 2}) + \ln (6^{3 x})$ .

$\ln (6^{2 x + 2}) + \ln (6^{3 x}) = \ln (1)$

Expand $\ln (6^{2 x + 2})$ by moving $2 x + 2$ outside the logarithm.

$(2 x + 2) \ln (6) + \ln (6^{3 x}) = \ln (1)$

Expand $\ln (6^{3 x})$ by moving $3 x$ outside the logarithm.

$(2 x + 2) \ln (6) + 3 x \ln (6) = \ln (1)$

Step 3

Simplify the left side.

Simplify $(2 x + 2) \ln (6) + 3 x \ln (6)$ .

Apply the distributive property.

$2 x \ln (6) + 2 \ln (6) + 3 x \ln (6) = \ln (1)$

Add $2 x \ln (6)$ and $3 x \ln (6)$ .

$5 x \ln (6) + 2 \ln (6) = \ln (1)$

Step 4

Simplify the right side.

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

$5 x \ln (6) + 2 \ln (6) = 0$

Step 5

Subtract $2 \ln (6)$ from both sides of the equation.

$5 x \ln (6) = - 2 \ln (6)$

Step 6

Divide each term in $5 x \ln (6) = - 2 \ln (6)$ by $5 \ln (6)$ and simplify.

Divide each term in $5 x \ln (6) = - 2 \ln (6)$ by $5 \ln (6)$ .

$\frac{5 x \ln (6)}{5 \ln (6)} = \frac{- 2 \ln (6)}{5 \ln (6)}$

Simplify the left side.

Cancel the common factor of $5$ .

Cancel the common factor.

$\frac{5 x \ln (6)}{5 \ln (6)} = \frac{- 2 \ln (6)}{5 \ln (6)}$

Rewrite the expression.

$\frac{x \ln (6)}{\ln (6)} = \frac{- 2 \ln (6)}{5 \ln (6)}$

Cancel the common factor of $\ln (6)$ .

Cancel the common factor.

$\frac{x \ln (6)}{\ln (6)} = \frac{- 2 \ln (6)}{5 \ln (6)}$

Divide $x$ by $1$ .

$x = \frac{- 2 \ln (6)}{5 \ln (6)}$

Simplify the right side.

Cancel the common factor of $\ln (6)$ .

Cancel the common factor.

$x = \frac{- 2 \ln (6)}{5 \ln (6)}$

Rewrite the expression.

$x = \frac{- 2}{5}$

Move the negative in front of the fraction.

$x = - \frac{2}{5}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{2}{5}$

Decimal Form:

$x = - 0.4$

Solve for x 6^(2x+2)*6^(3x)=1

Related Questions

Support LCMGCF.com by disabling your adblocker.