Solve for x 14^(x+1)=36

$14^{x + 1} = 36$

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

$\ln (14^{x + 1}) = \ln (36)$

Expand $\ln (14^{x + 1})$ by moving $x + 1$ outside the logarithm.

$(x + 1) \ln (14) = \ln (36)$

Simplify $(x + 1) \ln (14)$ .

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Apply the distributive property.

$x \ln (14) + 1 \ln (14) = \ln (36)$

Multiply $\ln (14)$ by $1$ .

$x \ln (14) + \ln (14) = \ln (36)$

Move all the terms containing a logarithm to the left side of the equation.

$x \ln (14) + \ln (14) - \ln (36) = 0$

Use the quotient property of logarithms, $\log_{b} (x) - \log_{b} (y) = \log_{b} (\frac{x}{y})$ .

$x \ln (14) + \ln (\frac{14}{36}) = 0$

Cancel the common factor of $14$ and $36$ .

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Factor $2$ out of $14$ .

$x \ln (14) + \ln (\frac{2 (7)}{36}) = 0$

Cancel the common factors.

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Factor $2$ out of $36$ .

$x \ln (14) + \ln (\frac{2 \cdot 7}{2 \cdot 18}) = 0$

Cancel the common factor.

$x \ln (14) + \ln (\frac{2 \cdot 7}{2 \cdot 18}) = 0$

Rewrite the expression.

$x \ln (14) + \ln (\frac{7}{18}) = 0$

Subtract $\ln (\frac{7}{18})$ from both sides of the equation.

$x \ln (14) = - \ln (\frac{7}{18})$

Divide each term by $\ln (14)$ and simplify.

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Divide each term in $x \ln (14) = - \ln (\frac{7}{18})$ by $\ln (14)$ .

$\frac{x \ln (14)}{\ln (14)} = \frac{- \ln (\frac{7}{18})}{\ln (14)}$

Cancel the common factor of $\ln (14)$ .

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Cancel the common factor.

$\frac{x \ln (14)}{\ln (14)} = \frac{- \ln (\frac{7}{18})}{\ln (14)}$

Divide $x$ by $1$ .

$x = \frac{- \ln (\frac{7}{18})}{\ln (14)}$

Move the negative in front of the fraction.

$x = - \frac{\ln (\frac{7}{18})}{\ln (14)}$

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{\ln (\frac{7}{18})}{\ln (14)}$

Decimal Form:

$x = 0.35787839 \dots$

Solve for x 14^(x+1)=36

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