Precalculus Examples

$e^{\frac{\ln (9) \ln (10)}{\ln (9) + \ln (10)}}$

Step 1

Simplify the numerator.

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Step 1.1

Rewrite $\ln (9)$ as $\ln (3^{2})$ .

$e^{\frac{\ln (3^{2}) \ln (10)}{\ln (9) + \ln (10)}}$

Step 1.2

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

$e^{\frac{2 \ln (3) \ln (10)}{\ln (9) + \ln (10)}}$

Step 1.3

Rewrite $\ln (10)$ as $\ln (2 (5))$ .

$e^{\frac{2 \ln (3) \ln (2 (5))}{\ln (9) + \ln (10)}}$

Step 1.4

Rewrite $\ln (2 (5))$ as $\ln (2) + \ln (5)$ .

$e^{\frac{2 \ln (3) (\ln (2) + \ln (5))}{\ln (9) + \ln (10)}}$

Step 2

Simplify the denominator.

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Step 2.1

Rewrite $\ln (9)$ as $\ln (3^{2})$ .

$e^{\frac{2 \ln (3) (\ln (2) + \ln (5))}{\ln (3^{2}) + \ln (10)}}$

Step 2.2

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

$e^{\frac{2 \ln (3) (\ln (2) + \ln (5))}{2 \ln (3) + \ln (10)}}$

Step 2.3

Rewrite $\ln (10)$ as $\ln (2 (5))$ .

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

Step 2.4

Rewrite $\ln (2 (5))$ as $\ln (2) + \ln (5)$ .

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

Step 3

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$3.07817127 \dots$