Precalculus Examples

${(1 + \frac{0.1}{12})}^{12 t} = 2$

Step 1

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

$\ln ({(1 + \frac{0.1}{12})}^{12 t}) = \ln (2)$

Step 2

Expand the left side.

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Step 2.1

Expand $\ln ({(1 + \frac{0.1}{12})}^{12 t})$ by moving $12 t$ outside the logarithm.

$12 t \ln (1 + \frac{0.1}{12}) = \ln (2)$

Step 2.2

Divide $0.1$ by $12$ .

$12 t \ln (1 + 0.008\overline{3}) = \ln (2)$

Step 2.3

Add $1$ and $0.008\overline{3}$ .

$12 t \ln (1.008\overline{3}) = \ln (2)$

Step 3

Divide each term in $12 t \ln (1.008\overline{3}) = \ln (2)$ by $12 \ln (1.008\overline{3})$ and simplify.

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Step 3.1

Divide each term in $12 t \ln (1.008\overline{3}) = \ln (2)$ by $12 \ln (1.008\overline{3})$ .

$\frac{12 t \ln (1.008\overline{3})}{12 \ln (1.008\overline{3})} = \frac{\ln (2)}{12 \ln (1.008\overline{3})}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Cancel the common factor of $12$ .

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Step 3.2.1.1

Cancel the common factor.

$\frac{12 t \ln (1.008\overline{3})}{12 \ln (1.008\overline{3})} = \frac{\ln (2)}{12 \ln (1.008\overline{3})}$

Step 3.2.1.2

Rewrite the expression.

$\frac{t \ln (1.008\overline{3})}{\ln (1.008\overline{3})} = \frac{\ln (2)}{12 \ln (1.008\overline{3})}$

Step 3.2.2

Cancel the common factor of $\ln (1.008\overline{3})$ .

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Step 3.2.2.1

Cancel the common factor.

$\frac{t \ln (1.008\overline{3})}{\ln (1.008\overline{3})} = \frac{\ln (2)}{12 \ln (1.008\overline{3})}$

Step 3.2.2.2

Divide $t$ by $1$ .

$t = \frac{\ln (2)}{12 \ln (1.008\overline{3})}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$t = \frac{\ln (2)}{12 \ln (1.008\overline{3})}$

Decimal Form:

$t = 6.96031299 \dots$