Precalculus Examples

$\frac{\ln (24) \ln (8)}{\log (24) + \log (8)}$

Step 1

Simplify the numerator.

Tap for more steps...

Step 1.1

Rewrite $\ln (24)$ as $\ln (2^{3} \cdot 3)$ .

$\frac{\ln (2^{3} \cdot 3) \ln (8)}{\log (24) + \log (8)}$

Step 1.2

Rewrite $\ln (2^{3} \cdot 3)$ as $\ln (2^{3}) + \ln (3)$ .

$\frac{(\ln (2^{3}) + \ln (3)) \ln (8)}{\log (24) + \log (8)}$

Step 1.3

Expand $\ln (2^{3})$ by moving $3$ outside the logarithm.

$\frac{(3 \ln (2) + \ln (3)) \ln (8)}{\log (24) + \log (8)}$

Step 1.4

Rewrite $\ln (8)$ as $\ln (2^{3})$ .

$\frac{(3 \ln (2) + \ln (3)) \ln (2^{3})}{\log (24) + \log (8)}$

Step 1.5

Expand $\ln (2^{3})$ by moving $3$ outside the logarithm.

$\frac{(3 \ln (2) + \ln (3)) \cdot 3 \ln (2)}{\log (24) + \log (8)}$

Step 2

Simplify the denominator.

Tap for more steps...

Step 2.1

Rewrite $\log (24)$ as $\log (2^{3} \cdot 3)$ .

$\frac{(3 \ln (2) + \ln (3)) \cdot 3 \ln (2)}{\log (2^{3} \cdot 3) + \log (8)}$

Step 2.2

Rewrite $\log (2^{3} \cdot 3)$ as $\log (2^{3}) + \log (3)$ .

$\frac{(3 \ln (2) + \ln (3)) \cdot 3 \ln (2)}{\log (2^{3}) + \log (3) + \log (8)}$

Step 2.3

Expand $\log (2^{3})$ by moving $3$ outside the logarithm.

$\frac{(3 \ln (2) + \ln (3)) \cdot 3 \ln (2)}{3 \log (2) + \log (3) + \log (8)}$

Step 2.4

Rewrite $\log (8)$ as $\log (2^{3})$ .

$\frac{(3 \ln (2) + \ln (3)) \cdot 3 \ln (2)}{3 \log (2) + \log (3) + \log (2^{3})}$

Step 2.5

Expand $\log (2^{3})$ by moving $3$ outside the logarithm.

$\frac{(3 \ln (2) + \ln (3)) \cdot 3 \ln (2)}{3 \log (2) + \log (3) + 3 \log (2)}$

Step 2.6

Add $3 \log (2)$ and $3 \log (2)$ .

$\frac{(3 \ln (2) + \ln (3)) \cdot 3 \ln (2)}{6 \log (2) + \log (3)}$

Step 3

Reorder.

Tap for more steps...

Step 3.1

Move $3$ to the left of $3 \ln (2) + \ln (3)$ .

$\frac{3 \cdot (3 \ln (2) + \ln (3)) \ln (2)}{6 \log (2) + \log (3)}$

Step 3.2

Reorder factors in $\frac{3 (3 \ln (2) + \ln (3)) \ln (2)}{6 \log (2) + \log (3)}$ .

$\frac{3 \ln (2) (3 \ln (2) + \ln (3))}{6 \log (2) + \log (3)}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$\frac{3 \ln (2) (3 \ln (2) + \ln (3))}{6 \log (2) + \log (3)}$

Decimal Form:

$2.89430806 \dots$