Precalculus Examples

$\frac{4 \ln (3)}{\ln (2)} + \frac{5 \ln (3)}{\ln (2)}$

Step 1

Combine the numerators over the common denominator.

$\frac{4 \ln (3) + 5 \ln (3)}{\ln (2)}$

Step 2

Simplify each term.

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Step 2.1

Simplify $4 \ln (3)$ by moving $4$ inside the logarithm.

$\frac{\ln (3^{4}) + 5 \ln (3)}{\ln (2)}$

Step 2.2

Raise $3$ to the power of $4$ .

$\frac{\ln (81) + 5 \ln (3)}{\ln (2)}$

Step 2.3

Simplify $5 \ln (3)$ by moving $5$ inside the logarithm.

$\frac{\ln (81) + \ln (3^{5})}{\ln (2)}$

Step 2.4

Raise $3$ to the power of $5$ .

$\frac{\ln (81) + \ln (243)}{\ln (2)}$

Step 3

Use the product property of logarithms, $\log_{b} (x) + \log_{b} (y) = \log_{b} (x y)$ .

$\frac{\ln (81 \cdot 243)}{\ln (2)}$

Step 4

Multiply $81$ by $243$ .

$\frac{\ln (19683)}{\ln (2)}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$\frac{\ln (19683)}{\ln (2)}$

Decimal Form:

$14.26466250 \dots$