Precalculus Examples

$\ln (e^{2.5}) - \ln (\sqrt{e})$

Step 1

Use the quotient property of logarithms, $\log_{b} (x) - \log_{b} (y) = \log_{b} (\frac{x}{y})$ .

$\ln (\frac{e^{2.5}}{\sqrt{e}})$

Step 2

Multiply $\frac{e^{2.5}}{\sqrt{e}}$ by $\frac{\sqrt{e}}{\sqrt{e}}$ .

$\ln (\frac{e^{2.5}}{\sqrt{e}} \cdot \frac{\sqrt{e}}{\sqrt{e}})$

Step 3

Combine and simplify the denominator.

Tap for more steps...

Step 3.1

Multiply $\frac{e^{2.5}}{\sqrt{e}}$ by $\frac{\sqrt{e}}{\sqrt{e}}$ .

$\ln (\frac{e^{2.5} \sqrt{e}}{\sqrt{e} \sqrt{e}})$

Step 3.2

Raise $\sqrt{e}$ to the power of $1$ .

$\ln (\frac{e^{2.5} \sqrt{e}}{{\sqrt{e}}^{1} \sqrt{e}})$

Step 3.3

Raise $\sqrt{e}$ to the power of $1$ .

$\ln (\frac{e^{2.5} \sqrt{e}}{{\sqrt{e}}^{1} {\sqrt{e}}^{1}})$

Step 3.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\ln (\frac{e^{2.5} \sqrt{e}}{{\sqrt{e}}^{1 + 1}})$

Step 3.5

Add $1$ and $1$ .

$\ln (\frac{e^{2.5} \sqrt{e}}{{\sqrt{e}}^{2}})$

Step 3.6

Rewrite ${\sqrt{e}}^{2}$ as $e$ .

Tap for more steps...

Step 3.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{e}$ as $e^{\frac{1}{2}}$ .

$\ln (\frac{e^{2.5} \sqrt{e}}{{(e^{\frac{1}{2}})}^{2}})$

Step 3.6.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\ln (\frac{e^{2.5} \sqrt{e}}{e^{\frac{1}{2} \cdot 2}})$

Step 3.6.3

Combine $\frac{1}{2}$ and $2$ .

$\ln (\frac{e^{2.5} \sqrt{e}}{e^{\frac{2}{2}}})$

Step 3.6.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 3.6.4.1

Cancel the common factor.

$\ln (\frac{e^{2.5} \sqrt{e}}{e^{\frac{2}{2}}})$

Step 3.6.4.2

Rewrite the expression.

$\ln (\frac{e^{2.5} \sqrt{e}}{e^{1}})$

Step 3.6.5

Simplify.

$\ln (\frac{e^{2.5} \sqrt{e}}{e})$

Step 4

Cancel the common factor of $e^{2.5}$ and $e$ .

Tap for more steps...

Step 4.1

Factor $e$ out of $e^{2.5} \sqrt{e}$ .

$\ln (\frac{e (e^{1.5} \sqrt{e})}{e})$

Step 4.2

Cancel the common factors.

Tap for more steps...

Step 4.2.1

Multiply by $1$ .

$\ln (\frac{e (e^{1.5} \sqrt{e})}{e \cdot 1})$

Step 4.2.2

Cancel the common factor.

$\ln (\frac{e (e^{1.5} \sqrt{e})}{e \cdot 1})$

Step 4.2.3

Rewrite the expression.

$\ln (\frac{e^{1.5} \sqrt{e}}{1})$

Step 4.2.4

Divide $e^{1.5} \sqrt{e}$ by $1$ .

$\ln (e^{1.5} \sqrt{e})$

Step 5

The result can be shown in multiple forms.

Exact Form:

$\ln (e^{1.5} \sqrt{e})$

Decimal Form:

$2$