Solve for x 6^(x+1)=4^(2x-1)

$6^{x + 1} = 4^{2 x - 1}$

Step 1

Take the log of both sides of the equation.

$\ln (6^{x + 1}) = \ln (4^{2 x - 1})$

Step 2

Expand $\ln (6^{x + 1})$ by moving $x + 1$ outside the logarithm.

$(x + 1) \ln (6) = \ln (4^{2 x - 1})$

Step 3

Expand $\ln (4^{2 x - 1})$ by moving $2 x - 1$ outside the logarithm.

$(x + 1) \ln (6) = (2 x - 1) \ln (4)$

Step 4

Solve the equation for $x$ .

Simplify $(x + 1) \ln (6)$ .

Rewrite.

$0 + 0 + (x + 1) \ln (6) = (2 x - 1) \ln (4)$

Simplify by adding zeros.

$(x + 1) \ln (6) = (2 x - 1) \ln (4)$

Apply the distributive property.

$x \ln (6) + 1 \ln (6) = (2 x - 1) \ln (4)$

Multiply $\ln (6)$ by $1$ .

$x \ln (6) + \ln (6) = (2 x - 1) \ln (4)$

Simplify $(2 x - 1) \ln (4)$ .

Apply the distributive property.

$x \ln (6) + \ln (6) = 2 x \ln (4) - 1 \ln (4)$

Rewrite $- 1 \ln (4)$ as $- \ln (4)$ .

$x \ln (6) + \ln (6) = 2 x \ln (4) - \ln (4)$

Subtract $2 x \ln (4)$ from both sides of the equation.

$x \ln (6) + \ln (6) - 2 x \ln (4) = - \ln (4)$

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

$x \ln (6) - 2 x \ln (4) = - \ln (4) - \ln (6)$

Factor $x$ out of $x \ln (6) - 2 x \ln (4)$ .

Factor $x$ out of $x \ln (6)$ .

$x (\ln (6)) - 2 x \ln (4) = - \ln (4) - \ln (6)$

Factor $x$ out of $- 2 x \ln (4)$ .

$x (\ln (6)) + x (- 2 \ln (4)) = - \ln (4) - \ln (6)$

Factor $x$ out of $x (\ln (6)) + x (- 2 \ln (4))$ .

$x (\ln (6) - 2 \ln (4)) = - \ln (4) - \ln (6)$

Divide each term in $x (\ln (6) - 2 \ln (4)) = - \ln (4) - \ln (6)$ by $\ln (6) - 2 \ln (4)$ and simplify.

Divide each term in $x (\ln (6) - 2 \ln (4)) = - \ln (4) - \ln (6)$ by $\ln (6) - 2 \ln (4)$ .

$\frac{x (\ln (6) - 2 \ln (4))}{\ln (6) - 2 \ln (4)} = \frac{- \ln (4)}{\ln (6) - 2 \ln (4)} + \frac{- \ln (6)}{\ln (6) - 2 \ln (4)}$

Simplify the left side.

Cancel the common factor of $\ln (6) - 2 \ln (4)$ .

Cancel the common factor.

$\frac{x (\ln (6) - 2 \ln (4))}{\ln (6) - 2 \ln (4)} = \frac{- \ln (4)}{\ln (6) - 2 \ln (4)} + \frac{- \ln (6)}{\ln (6) - 2 \ln (4)}$

Divide $x$ by $1$ .

$x = \frac{- \ln (4)}{\ln (6) - 2 \ln (4)} + \frac{- \ln (6)}{\ln (6) - 2 \ln (4)}$

Simplify the right side.

Combine the numerators over the common denominator.

$x = \frac{- \ln (4) - \ln (6)}{\ln (6) - 2 \ln (4)}$

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

$x = \frac{- \ln (4) - \ln (6)}{\ln (6) - 2 \ln (4)}$

Factor $- 1$ out of $- \ln (6)$ .

$x = \frac{- \ln (4) - \ln (6)}{\ln (6) - 2 \ln (4)}$

Factor $- 1$ out of $- \ln (4) - \ln (6)$ .

$x = \frac{- (\ln (4) + \ln (6))}{\ln (6) - 2 \ln (4)}$

Simplify the expression.

Rewrite $- (\ln (4) + \ln (6))$ as $- 1 (\ln (4) + \ln (6))$ .

$x = \frac{- 1 (\ln (4) + \ln (6))}{\ln (6) - 2 \ln (4)}$

Move the negative in front of the fraction.

$x = - \frac{\ln (4) + \ln (6)}{\ln (6) - 2 \ln (4)}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{\ln (4) + \ln (6)}{\ln (6) - 2 \ln (4)}$

Decimal Form:

$x = 3.24017031 \dots$

Solve for x 6^(x+1)=4^(2x-1)

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