Solve for x 5.9^(x+1)=15

${5.9}^{x + 1} = 15$

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

$\ln ({5.9}^{x + 1}) = \ln (15)$

Expand $\ln ({5.9}^{x + 1})$ by moving $x + 1$ outside the logarithm.

$(x + 1) \ln (5.9) = \ln (15)$

Simplify the left side.

Simplify $(x + 1) \ln (5.9)$ .

Apply the distributive property.

$x \ln (5.9) + 1 \ln (5.9) = \ln (15)$

Multiply $\ln (5.9)$ by $1$ .

$x \ln (5.9) + \ln (5.9) = \ln (15)$

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

$x \ln (5.9) + \ln (5.9) - \ln (15) = 0$

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

$x \ln (5.9) + \ln (\frac{5.9}{15}) = 0$

Divide $5.9$ by $15$ .

$x \ln (5.9) + \ln (0.39\overline{3}) = 0$

Subtract $\ln (0.39\overline{3})$ from both sides of the equation.

$x \ln (5.9) = - \ln (0.39\overline{3})$

Divide each term in $x \ln (5.9) = - \ln (0.39\overline{3})$ by $\ln (5.9)$ and simplify.

Divide each term in $x \ln (5.9) = - \ln (0.39\overline{3})$ by $\ln (5.9)$ .

$\frac{x \ln (5.9)}{\ln (5.9)} = \frac{- \ln (0.39\overline{3})}{\ln (5.9)}$

Simplify the left side.

Cancel the common factor of $\ln (5.9)$ .

Cancel the common factor.

$\frac{x \ln (5.9)}{\ln (5.9)} = \frac{- \ln (0.39\overline{3})}{\ln (5.9)}$

Divide $x$ by $1$ .

$x = \frac{- \ln (0.39\overline{3})}{\ln (5.9)}$

Simplify the right side.

Move the negative in front of the fraction.

$x = - \frac{\ln (0.39\overline{3})}{\ln (5.9)}$

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{\ln (0.39\overline{3})}{\ln (5.9)}$

Decimal Form:

$x = 0.52570304 \dots$