Solve for t natural log of 0.5/2.6=(-( natural log of 2)/5730)t

$\ln (\frac{0.5}{2.6}) = (- \frac{\ln (2)}{5730}) t$

Rewrite the equation as $- \frac{\ln (2)}{5730} t = \ln (\frac{0.5}{2.6})$ .

$- \frac{\ln (2)}{5730} t = \ln (\frac{0.5}{2.6})$

Simplify the left side.

Combine $t$ and $\frac{\ln (2)}{5730}$ .

$- \frac{t \ln (2)}{5730} = \ln (\frac{0.5}{2.6})$

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

$- \frac{t \ln (2)}{5730} - \ln (\frac{0.5}{2.6}) = 0$

Divide $0.5$ by $2.6$ .

$- \frac{t \ln (2)}{5730} - \ln (0.1\overline{923076}) = 0$

Add $\ln (0.1\overline{923076})$ to both sides of the equation.

$- \frac{t \ln (2)}{5730} = \ln (0.1\overline{923076})$

Multiply both sides of the equation by $- 5730$ .

$- 5730 (- \frac{t \ln (2)}{5730}) = - 5730 \ln (0.1\overline{923076})$

Simplify the left side.

Simplify $- 5730 (- \frac{t \ln (2)}{5730})$ .

Cancel the common factor of $5730$ .

Move the leading negative in $- \frac{t \ln (2)}{5730}$ into the numerator.

$- 5730 \frac{- t \ln (2)}{5730} = - 5730 \ln (0.1\overline{923076})$

Factor $5730$ out of $- 5730$ .

$5730 (- 1) \frac{- t \ln (2)}{5730} = - 5730 \ln (0.1\overline{923076})$

Cancel the common factor.

$5730 \cdot - 1 \frac{- t \ln (2)}{5730} = - 5730 \ln (0.1\overline{923076})$

Rewrite the expression.

$- (- t \ln (2)) = - 5730 \ln (0.1\overline{923076})$

Multiply.

Multiply $- 1$ by $- 1$ .

$1 (t \ln (2)) = - 5730 \ln (0.1\overline{923076})$

Multiply $t$ by $1$ .

$t \ln (2) = - 5730 \ln (0.1\overline{923076})$

Divide each term in $t \ln (2) = - 5730 \ln (0.1\overline{923076})$ by $\ln (2)$ and simplify.

Divide each term in $t \ln (2) = - 5730 \ln (0.1\overline{923076})$ by $\ln (2)$ .

$\frac{t \ln (2)}{\ln (2)} = \frac{- 5730 \ln (0.1\overline{923076})}{\ln (2)}$

Simplify the left side.

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

Cancel the common factor.

$\frac{t \ln (2)}{\ln (2)} = \frac{- 5730 \ln (0.1\overline{923076})}{\ln (2)}$

Divide $t$ by $1$ .

$t = \frac{- 5730 \ln (0.1\overline{923076})}{\ln (2)}$

Simplify the right side.

Move the negative in front of the fraction.

$t = - \frac{5730 \ln (0.1\overline{923076})}{\ln (2)}$

The result can be shown in multiple forms.

Exact Form:

$t = - \frac{5730 \ln (0.1\overline{923076})}{\ln (2)}$

Decimal Form:

$t = 13628.87160124 \dots$

Solve for t natural log of 0.5/2.6=(-( natural log of 2)/5730)t

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