Find the Exact Value 1/( log base 2 of 14)+1/( log base 7 of 14)

$\frac{1}{\log_{2} (14)} + \frac{1}{\log_{7} (14)}$

To write $\frac{1}{\log_{2} (14)}$ as a fraction with a common denominator, multiply by $\frac{\log_{7} (14)}{\log_{7} (14)}$ .

$\frac{1}{\log_{2} (14)} \cdot \frac{\log_{7} (14)}{\log_{7} (14)} + \frac{1}{\log_{7} (14)}$

To write $\frac{1}{\log_{7} (14)}$ as a fraction with a common denominator, multiply by $\frac{\log_{2} (14)}{\log_{2} (14)}$ .

$\frac{1}{\log_{2} (14)} \cdot \frac{\log_{7} (14)}{\log_{7} (14)} + \frac{1}{\log_{7} (14)} \cdot \frac{\log_{2} (14)}{\log_{2} (14)}$

Write each expression with a common denominator of $\log_{2} (14) \log_{7} (14)$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{1}{\log_{2} (14)}$ by $\frac{\log_{7} (14)}{\log_{7} (14)}$ .

$\frac{\log_{7} (14)}{\log_{2} (14) \log_{7} (14)} + \frac{1}{\log_{7} (14)} \cdot \frac{\log_{2} (14)}{\log_{2} (14)}$

Multiply $\frac{1}{\log_{7} (14)}$ by $\frac{\log_{2} (14)}{\log_{2} (14)}$ .

$\frac{\log_{7} (14)}{\log_{2} (14) \log_{7} (14)} + \frac{\log_{2} (14)}{\log_{7} (14) \log_{2} (14)}$

Reorder the factors of $\log_{7} (14) \log_{2} (14)$ .

$\frac{\log_{7} (14)}{\log_{2} (14) \log_{7} (14)} + \frac{\log_{2} (14)}{\log_{2} (14) \log_{7} (14)}$

Combine the numerators over the common denominator.

$\frac{\log_{7} (14) + \log_{2} (14)}{\log_{2} (14) \log_{7} (14)}$

The result can be shown in multiple forms.

Exact Form:

$\frac{\log_{7} (14) + \log_{2} (14)}{\log_{2} (14) \log_{7} (14)}$

Decimal Form:

$1$