Precalculus Examples

$4^{0.6 \log_{4} (\sqrt{8})} - \ln (e^{\ln (6)})$

Step 1

Simplify $0.6 \log_{4} (\sqrt{8})$ by moving $0.6$ inside the logarithm.

$4^{\log_{4} ({\sqrt{8}}^{0.6})} - \ln (e^{\ln (6)})$

Step 2

Exponentiation and log are inverse functions.

${\sqrt{8}}^{0.6} - \ln (e^{\ln (6)})$

Step 3

Rewrite $8$ as $2^{2} \cdot 2$ .

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Step 3.1

Factor $4$ out of $8$ .

${\sqrt{4 (2)}}^{0.6} - \ln (e^{\ln (6)})$

Step 3.2

Rewrite $4$ as $2^{2}$ .

${\sqrt{2^{2} \cdot 2}}^{0.6} - \ln (e^{\ln (6)})$

Step 4

Pull terms out from under the radical.

${(2 \sqrt{2})}^{0.6} - \ln (e^{\ln (6)})$

Step 5

Apply the product rule to $2 \sqrt{2}$ .

$2^{0.6} {\sqrt{2}}^{0.6} - \ln (e^{\ln (6)})$

Step 6

Raise $2$ to the power of $0.6$ .

$1.51571656 {\sqrt{2}}^{0.6} - \ln (e^{\ln (6)})$

Step 7

Multiply $1.51571656$ by ${\sqrt{2}}^{0.6}$ .

$1.86606598 - \ln (e^{\ln (6)})$

Step 8

Use logarithm rules to move $\ln (6)$ out of the exponent.

$1.86606598 - (\ln (6) \ln (e))$

Step 9

The natural logarithm of $e$ is $1$ .

$1.86606598 - (\ln (6) \cdot 1)$

Step 10

Multiply $\ln (6)$ by $1$ .

$1.86606598 - \ln (6)$

Step 11

The result can be shown in multiple forms.

Exact Form:

$1.86606598 - \ln (6)$

Decimal Form:

$0.07430651 \dots$