Solve for x 7^(2x+1)=3^(x+2)

$7^{2 x + 1} = 3^{x + 2}$

Step 1

Take the log of both sides of the equation.

$\ln (7^{2 x + 1}) = \ln (3^{x + 2})$

Step 2

Expand $\ln (7^{2 x + 1})$ by moving $2 x + 1$ outside the logarithm.

$(2 x + 1) \ln (7) = \ln (3^{x + 2})$

Step 3

Expand $\ln (3^{x + 2})$ by moving $x + 2$ outside the logarithm.

$(2 x + 1) \ln (7) = (x + 2) \ln (3)$

Step 4

Solve the equation for $x$ .

Simplify $(2 x + 1) \ln (7)$ .

Rewrite.

$0 + 0 + (2 x + 1) \ln (7) = (x + 2) \ln (3)$

Simplify by adding zeros.

$(2 x + 1) \ln (7) = (x + 2) \ln (3)$

Apply the distributive property.

$2 x \ln (7) + 1 \ln (7) = (x + 2) \ln (3)$

Multiply $\ln (7)$ by $1$ .

$2 x \ln (7) + \ln (7) = (x + 2) \ln (3)$

Apply the distributive property.

$2 x \ln (7) + \ln (7) = x \ln (3) + 2 \ln (3)$

Subtract $x \ln (3)$ from both sides of the equation.

$2 x \ln (7) + \ln (7) - x \ln (3) = 2 \ln (3)$

Subtract $\ln (7)$ from both sides of the equation.

$2 x \ln (7) - x \ln (3) = 2 \ln (3) - \ln (7)$

Factor $x$ out of $2 x \ln (7) - x \ln (3)$ .

Factor $x$ out of $2 x \ln (7)$ .

$x (2 \ln (7)) - x \ln (3) = 2 \ln (3) - \ln (7)$

Factor $x$ out of $- x \ln (3)$ .

$x (2 \ln (7)) + x (- 1 \ln (3)) = 2 \ln (3) - \ln (7)$

Factor $x$ out of $x (2 \ln (7)) + x (- 1 \ln (3))$ .

$x (2 \ln (7) - 1 \ln (3)) = 2 \ln (3) - \ln (7)$

Rewrite $- 1 \ln (3)$ as $- \ln (3)$ .

$x (2 \ln (7) - \ln (3)) = 2 \ln (3) - \ln (7)$

Divide each term in $x (2 \ln (7) - \ln (3)) = 2 \ln (3) - \ln (7)$ by $2 \ln (7) - \ln (3)$ and simplify.

Divide each term in $x (2 \ln (7) - \ln (3)) = 2 \ln (3) - \ln (7)$ by $2 \ln (7) - \ln (3)$ .

$\frac{x (2 \ln (7) - \ln (3))}{2 \ln (7) - \ln (3)} = \frac{2 \ln (3)}{2 \ln (7) - \ln (3)} + \frac{- \ln (7)}{2 \ln (7) - \ln (3)}$

Simplify the left side.

Cancel the common factor of $2 \ln (7) - \ln (3)$ .

Cancel the common factor.

$\frac{x (2 \ln (7) - \ln (3))}{2 \ln (7) - \ln (3)} = \frac{2 \ln (3)}{2 \ln (7) - \ln (3)} + \frac{- \ln (7)}{2 \ln (7) - \ln (3)}$

Divide $x$ by $1$ .

$x = \frac{2 \ln (3)}{2 \ln (7) - \ln (3)} + \frac{- \ln (7)}{2 \ln (7) - \ln (3)}$

Simplify the right side.

Combine the numerators over the common denominator.

$x = \frac{2 \ln (3) - \ln (7)}{2 \ln (7) - \ln (3)}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = \frac{2 \ln (3) - \ln (7)}{2 \ln (7) - \ln (3)}$

Decimal Form:

$x = 0.08997340 \dots$

Solve for x 7^(2x+1)=3^(x+2)

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