Basic Math Examples

$\ln (\frac{0.1}{0.6}) = \frac{\ln (\frac{0.56}{0.6})}{30} t$

Step 1

Rewrite the equation as $\frac{\ln (\frac{0.56}{0.6})}{30} t = \ln (\frac{0.1}{0.6})$ .

$\frac{\ln (\frac{0.56}{0.6})}{30} t = \ln (\frac{0.1}{0.6})$

Step 2

Simplify the left side.

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Step 2.1

Combine $\frac{\ln (\frac{0.56}{0.6})}{30}$ and $t$ .

$\frac{\ln (\frac{0.56}{0.6}) t}{30} = \ln (\frac{0.1}{0.6})$

Step 3

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

$\frac{\ln (\frac{0.56}{0.6}) t}{30} - \ln (\frac{0.1}{0.6}) = 0$

Step 4

Simplify each term.

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Step 4.1

Divide $0.56$ by $0.6$ .

$\frac{\ln (0.9\overline{3}) t}{30} - \ln (\frac{0.1}{0.6}) = 0$

Step 4.2

Divide $0.1$ by $0.6$ .

$\frac{\ln (0.9\overline{3}) t}{30} - \ln (0.1\overline{6}) = 0$

Step 5

Reorder factors in $\frac{\ln (0.9\overline{3}) t}{30} - \ln (0.1\overline{6})$ .

$\frac{t \ln (0.9\overline{3})}{30} - \ln (0.1\overline{6}) = 0$

Step 6

Add $\ln (0.1\overline{6})$ to both sides of the equation.

$\frac{t \ln (0.9\overline{3})}{30} = \ln (0.1\overline{6})$

Step 7

Multiply both sides of the equation by $30$ .

$30 \frac{t \ln (0.9\overline{3})}{30} = 30 \ln (0.1\overline{6})$

Step 8

Simplify the left side.

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Step 8.1

Cancel the common factor of $30$ .

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Step 8.1.1

Cancel the common factor.

$30 \frac{t \ln (0.9\overline{3})}{30} = 30 \ln (0.1\overline{6})$

Step 8.1.2

Rewrite the expression.

$t \ln (0.9\overline{3}) = 30 \ln (0.1\overline{6})$

Step 9

Divide each term in $t \ln (0.9\overline{3}) = 30 \ln (0.1\overline{6})$ by $\ln (0.9\overline{3})$ and simplify.

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Step 9.1

Divide each term in $t \ln (0.9\overline{3}) = 30 \ln (0.1\overline{6})$ by $\ln (0.9\overline{3})$ .

$\frac{t \ln (0.9\overline{3})}{\ln (0.9\overline{3})} = \frac{30 \ln (0.1\overline{6})}{\ln (0.9\overline{3})}$

Step 9.2

Simplify the left side.

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Step 9.2.1

Cancel the common factor of $\ln (0.9\overline{3})$ .

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Step 9.2.1.1

Cancel the common factor.

$\frac{t \ln (0.9\overline{3})}{\ln (0.9\overline{3})} = \frac{30 \ln (0.1\overline{6})}{\ln (0.9\overline{3})}$

Step 9.2.1.2

Divide $t$ by $1$ .

$t = \frac{30 \ln (0.1\overline{6})}{\ln (0.9\overline{3})}$

Step 10

The result can be shown in multiple forms.

Exact Form:

$t = \frac{30 \ln (0.1\overline{6})}{\ln (0.9\overline{3})}$

Decimal Form:

$t = 779.10634705 \dots$