Solve for x 3^(x+4)=7^(x-1)

$3^{x + 4} = 7^{x - 1}$

Step 1

Take the log of both sides of the equation.

$\ln (3^{x + 4}) = \ln (7^{x - 1})$

Step 2

Expand $\ln (3^{x + 4})$ by moving $x + 4$ outside the logarithm.

$(x + 4) \ln (3) = \ln (7^{x - 1})$

Step 3

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

$(x + 4) \ln (3) = (x - 1) \ln (7)$

Step 4

Solve the equation for $x$ .

Simplify $(x + 4) \ln (3)$ .

Rewrite.

$0 + 0 + (x + 4) \ln (3) = (x - 1) \ln (7)$

Simplify by adding zeros.

$(x + 4) \ln (3) = (x - 1) \ln (7)$

Apply the distributive property.

$x \ln (3) + 4 \ln (3) = (x - 1) \ln (7)$

Simplify $(x - 1) \ln (7)$ .

Apply the distributive property.

$x \ln (3) + 4 \ln (3) = x \ln (7) - 1 \ln (7)$

Rewrite $- 1 \ln (7)$ as $- \ln (7)$ .

$x \ln (3) + 4 \ln (3) = x \ln (7) - \ln (7)$

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

$x \ln (3) + 4 \ln (3) - x \ln (7) = - \ln (7)$

Subtract $4 \ln (3)$ from both sides of the equation.

$x \ln (3) - x \ln (7) = - \ln (7) - 4 \ln (3)$

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

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

$x (\ln (3)) - x \ln (7) = - \ln (7) - 4 \ln (3)$

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

$x (\ln (3)) + x (- 1 \ln (7)) = - \ln (7) - 4 \ln (3)$

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

$x (\ln (3) - 1 \ln (7)) = - \ln (7) - 4 \ln (3)$

Rewrite $- 1 \ln (7)$ as $- \ln (7)$ .

$x (\ln (3) - \ln (7)) = - \ln (7) - 4 \ln (3)$

Divide each term in $x (\ln (3) - \ln (7)) = - \ln (7) - 4 \ln (3)$ by $\ln (3) - \ln (7)$ and simplify.

Divide each term in $x (\ln (3) - \ln (7)) = - \ln (7) - 4 \ln (3)$ by $\ln (3) - \ln (7)$ .

$\frac{x (\ln (3) - \ln (7))}{\ln (3) - \ln (7)} = \frac{- \ln (7)}{\ln (3) - \ln (7)} + \frac{- 4 \ln (3)}{\ln (3) - \ln (7)}$

Simplify the left side.

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

Cancel the common factor.

$\frac{x (\ln (3) - \ln (7))}{\ln (3) - \ln (7)} = \frac{- \ln (7)}{\ln (3) - \ln (7)} + \frac{- 4 \ln (3)}{\ln (3) - \ln (7)}$

Divide $x$ by $1$ .

$x = \frac{- \ln (7)}{\ln (3) - \ln (7)} + \frac{- 4 \ln (3)}{\ln (3) - \ln (7)}$

Simplify the right side.

Combine the numerators over the common denominator.

$x = \frac{- \ln (7) - 4 \ln (3)}{\ln (3) - \ln (7)}$

Factor $- 1$ out of $- \ln (7)$ .

$x = \frac{- \ln (7) - 4 \ln (3)}{\ln (3) - \ln (7)}$

Factor $- 1$ out of $- 4 \ln (3)$ .

$x = \frac{- \ln (7) - (4 \ln (3))}{\ln (3) - \ln (7)}$

Factor $- 1$ out of $- \ln (7) - (4 \ln (3))$ .

$x = \frac{- (\ln (7) + 4 \ln (3))}{\ln (3) - \ln (7)}$

Simplify the expression.

Rewrite $- (\ln (7) + 4 \ln (3))$ as $- 1 (\ln (7) + 4 \ln (3))$ .

$x = \frac{- 1 (\ln (7) + 4 \ln (3))}{\ln (3) - \ln (7)}$

Move the negative in front of the fraction.

$x = - \frac{\ln (7) + 4 \ln (3)}{\ln (3) - \ln (7)}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{\ln (7) + 4 \ln (3)}{\ln (3) - \ln (7)}$

Decimal Form:

$x = 7.48303471 \dots$

Solve for x 3^(x+4)=7^(x-1)

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