Algebra Examples

\log_{11} (121) + \log_{2} (8^{3}) - 3 \ln (e^{4})

$\log_{11} (121) + \log_{2} (8^{3}) - 3 \ln (e^{4})$

Step 1

Simplify each term.

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Step 1.1

Logarithm base

11

$11$ of

121

$121$ is

2

$2$ .

2 + \log_{2} (8^{3}) - 3 \ln (e^{4})

$2 + \log_{2} (8^{3}) - 3 \ln (e^{4})$

Step 1.2

Raise

8

$8$ to the power of

3

$3$ .

2 + \log_{2} (512) - 3 \ln (e^{4})

$2 + \log_{2} (512) - 3 \ln (e^{4})$

Step 1.3

Logarithm base

2

$2$ of

512

$512$ is

9

$9$ .

2 + 9 - 3 \ln (e^{4})

$2 + 9 - 3 \ln (e^{4})$

Step 1.4

Use logarithm rules to move

4

$4$ out of the exponent.

2 + 9 - 3 (4 \ln (e))

$2 + 9 - 3 (4 \ln (e))$

Step 1.5

The natural logarithm of

e

$e$ is

1

$1$ .

2 + 9 - 3 (4 \cdot 1)

$2 + 9 - 3 (4 \cdot 1)$

Step 1.6

Multiply

4

$4$ by

1

$1$ .

2 + 9 - 3 \cdot 4

$2 + 9 - 3 \cdot 4$

Step 1.7

Multiply

- 3

$- 3$ by

4

$4$ .

2 + 9 - 12

$2 + 9 - 12$

2 + 9 - 12

$2 + 9 - 12$

Step 2

Simplify by adding and subtracting.

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Step 2.1

Add

2

$2$ and

9

$9$ .

11 - 12

$11 - 12$

Step 2.2

Subtract

12

$12$ from

11

$11$ .

- 1

$- 1$

- 1

$- 1$

[\begin{array}{c} x^{2} & \frac{1}{2} \\ \sqrt{π} & \int x d x \end{array}]

$[\begin{array}{c} x^{2} & \frac{1}{2} \\ \sqrt{π} & \int x d x \end{array}]$