Basic Math Examples

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

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

Step 1

Rewrite the equation as

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

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

\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}

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

t

$t$ .

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

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

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

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

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

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

Step 4

Simplify each term.

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Step 4.1

Divide

0.56

$0.56$ by

0.6

$0.6$ .

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

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

Step 4.2

Divide

0.1

$0.1$ by

0.6

$0.6$ .

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

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

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

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

Step 5

Reorder factors in

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

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

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

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

Step 6

Add

\ln (0.1\overline{6})

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

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

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

Step 7

Multiply both sides of the equation by

30

$30$ .

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

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

Step 8

Simplify the left side.

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Step 8.1

Cancel the common factor of

30

$30$ .

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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})

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

Step 8.1.2

Rewrite the expression.

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

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

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

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

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

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

Step 9

Divide each term in

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

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

\ln (0.9\overline{3})

$\ln (0.9\overline{3})$ 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})

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

\ln (0.9\overline{3})

$\ln (0.9\overline{3})$ .

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

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

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})

$\ln (0.9\overline{3})$ .

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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})}

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

Step 9.2.1.2

Divide

t

$t$ by

1

$1$ .

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