Finite Math Examples

${(\sqrt{3})}^{7} + {(\sqrt{3})}^{5} + {(\sqrt{3})}^{3} + 42 (\sqrt{3}) = 3^{k}$

Step 1

Rewrite the equation as $3^{k} = {\sqrt{3}}^{7} + {\sqrt{3}}^{5} + {\sqrt{3}}^{3} + 42 \sqrt{3}$ .

$3^{k} = {\sqrt{3}}^{7} + {\sqrt{3}}^{5} + {\sqrt{3}}^{3} + 42 \sqrt{3}$

Step 2

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

$\ln (3^{k}) = \ln (81 \sqrt{3})$

Step 3

Expand $\ln (3^{k})$ by moving $k$ outside the logarithm.

$k \ln (3) = \ln (81 \sqrt{3})$

Step 4

Divide each term in $k \ln (3) = \ln (81 \sqrt{3})$ by $\ln (3)$ and simplify.

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Step 4.1

Divide each term in $k \ln (3) = \ln (81 \sqrt{3})$ by $\ln (3)$ .

$\frac{k \ln (3)}{\ln (3)} = \frac{\ln (81 \sqrt{3})}{\ln (3)}$

Step 4.2

Simplify the left side.

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Step 4.2.1

Cancel the common factor of $\ln (3)$ .

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Step 4.2.1.1

Cancel the common factor.

$\frac{k \ln (3)}{\ln (3)} = \frac{\ln (81 \sqrt{3})}{\ln (3)}$

Step 4.2.1.2

Divide $k$ by $1$ .

$k = \frac{\ln (81 \sqrt{3})}{\ln (3)}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$k = \frac{\ln (81 \sqrt{3})}{\ln (3)}$

Decimal Form:

$k = 4.5$