Finite Math Examples

${1.684}^{12 t} = 3$

Step 1

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

$\ln ({1.684}^{12 t}) = \ln (3)$

Step 2

Expand $\ln ({1.684}^{12 t})$ by moving $12 t$ outside the logarithm.

$12 t \ln (1.684) = \ln (3)$

Step 3

Divide each term in $12 t \ln (1.684) = \ln (3)$ by $12 \ln (1.684)$ and simplify.

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Step 3.1

Divide each term in $12 t \ln (1.684) = \ln (3)$ by $12 \ln (1.684)$ .

$\frac{12 t \ln (1.684)}{12 \ln (1.684)} = \frac{\ln (3)}{12 \ln (1.684)}$

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.

$\frac{12 t \ln (1.684)}{12 \ln (1.684)} = \frac{\ln (3)}{12 \ln (1.684)}$

Step 3.2.1.2

Rewrite the expression.

$\frac{t \ln (1.684)}{\ln (1.684)} = \frac{\ln (3)}{12 \ln (1.684)}$

Step 3.2.2

Cancel the common factor of $\ln (1.684)$ .

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Step 3.2.2.1

Cancel the common factor.

$\frac{t \ln (1.684)}{\ln (1.684)} = \frac{\ln (3)}{12 \ln (1.684)}$

Step 3.2.2.2

Divide $t$ by $1$ .

$t = \frac{\ln (3)}{12 \ln (1.684)}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$t = \frac{\ln (3)}{12 \ln (1.684)}$

Decimal Form:

$t = 0.17566377 \dots$