Evaluate 5^(x+1)=106

$5^{x + 1} = 106$

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

$\ln (5^{x + 1}) = \ln (106)$

Expand $\ln (5^{x + 1})$ by moving $x + 1$ outside the logarithm.

$(x + 1) \ln (5) = \ln (106)$

Simplify the left side.

Simplify $(x + 1) \ln (5)$ .

Apply the distributive property.

$x \ln (5) + 1 \ln (5) = \ln (106)$

Multiply $\ln (5)$ by $1$ .

$x \ln (5) + \ln (5) = \ln (106)$

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

$x \ln (5) + \ln (5) - \ln (106) = 0$

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

$x \ln (5) + \ln (\frac{5}{106}) = 0$

Subtract $\ln (\frac{5}{106})$ from both sides of the equation.

$x \ln (5) = - \ln (\frac{5}{106})$

Divide each term in $x \ln (5) = - \ln (\frac{5}{106})$ by $\ln (5)$ and simplify.

Divide each term in $x \ln (5) = - \ln (\frac{5}{106})$ by $\ln (5)$ .

$\frac{x \ln (5)}{\ln (5)} = \frac{- \ln (\frac{5}{106})}{\ln (5)}$

Simplify the left side.

Cancel the common factor of $\ln (5)$ .

Cancel the common factor.

$\frac{x \ln (5)}{\ln (5)} = \frac{- \ln (\frac{5}{106})}{\ln (5)}$

Divide $x$ by $1$ .

$x = \frac{- \ln (\frac{5}{106})}{\ln (5)}$

Simplify the right side.

Move the negative in front of the fraction.

$x = - \frac{\ln (\frac{5}{106})}{\ln (5)}$

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{\ln (\frac{5}{106})}{\ln (5)}$

Decimal Form:

$x = 1.89755762 \dots$