Find the Exact Value 1/( log base 2 of 30)+1/( log base 3 of 30)+1/( log base 5 of 30)

$\frac{1}{\log_{2} (30)} + \frac{1}{\log_{3} (30)} + \frac{1}{\log_{5} (30)}$

Step 1

To write $\frac{1}{\log_{2} (30)}$ as a fraction with a common denominator, multiply by $\frac{\log_{3} (30)}{\log_{3} (30)}$ .

$\frac{1}{\log_{2} (30)} \cdot \frac{\log_{3} (30)}{\log_{3} (30)} + \frac{1}{\log_{3} (30)} + \frac{1}{\log_{5} (30)}$

Step 2

To write $\frac{1}{\log_{3} (30)}$ as a fraction with a common denominator, multiply by $\frac{\log_{2} (30)}{\log_{2} (30)}$ .

$\frac{1}{\log_{2} (30)} \cdot \frac{\log_{3} (30)}{\log_{3} (30)} + \frac{1}{\log_{3} (30)} \cdot \frac{\log_{2} (30)}{\log_{2} (30)} + \frac{1}{\log_{5} (30)}$

Step 3

Write each expression with a common denominator of $\log_{2} (30) \log_{3} (30)$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{1}{\log_{2} (30)}$ by $\frac{\log_{3} (30)}{\log_{3} (30)}$ .

$\frac{\log_{3} (30)}{\log_{2} (30) \log_{3} (30)} + \frac{1}{\log_{3} (30)} \cdot \frac{\log_{2} (30)}{\log_{2} (30)} + \frac{1}{\log_{5} (30)}$

Multiply $\frac{1}{\log_{3} (30)}$ by $\frac{\log_{2} (30)}{\log_{2} (30)}$ .

$\frac{\log_{3} (30)}{\log_{2} (30) \log_{3} (30)} + \frac{\log_{2} (30)}{\log_{3} (30) \log_{2} (30)} + \frac{1}{\log_{5} (30)}$

Reorder the factors of $\log_{3} (30) \log_{2} (30)$ .

$\frac{\log_{3} (30)}{\log_{2} (30) \log_{3} (30)} + \frac{\log_{2} (30)}{\log_{2} (30) \log_{3} (30)} + \frac{1}{\log_{5} (30)}$

Step 4

Combine the numerators over the common denominator.

$\frac{\log_{3} (30) + \log_{2} (30)}{\log_{2} (30) \log_{3} (30)} + \frac{1}{\log_{5} (30)}$

Step 5

To write $\frac{\log_{3} (30) + \log_{2} (30)}{\log_{2} (30) \log_{3} (30)}$ as a fraction with a common denominator, multiply by $\frac{\log_{5} (30)}{\log_{5} (30)}$ .

$\frac{\log_{3} (30) + \log_{2} (30)}{\log_{2} (30) \log_{3} (30)} \cdot \frac{\log_{5} (30)}{\log_{5} (30)} + \frac{1}{\log_{5} (30)}$

Step 6

To write $\frac{1}{\log_{5} (30)}$ as a fraction with a common denominator, multiply by $\frac{\log_{2} (30) \log_{3} (30)}{\log_{2} (30) \log_{3} (30)}$ .

$\frac{\log_{3} (30) + \log_{2} (30)}{\log_{2} (30) \log_{3} (30)} \cdot \frac{\log_{5} (30)}{\log_{5} (30)} + \frac{1}{\log_{5} (30)} \cdot \frac{\log_{2} (30) \log_{3} (30)}{\log_{2} (30) \log_{3} (30)}$

Step 7

Write each expression with a common denominator of $\log_{2} (30) \log_{3} (30) \log_{5} (30)$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{\log_{3} (30) + \log_{2} (30)}{\log_{2} (30) \log_{3} (30)}$ by $\frac{\log_{5} (30)}{\log_{5} (30)}$ .

$\frac{(\log_{3} (30) + \log_{2} (30)) \log_{5} (30)}{\log_{2} (30) \log_{3} (30) \log_{5} (30)} + \frac{1}{\log_{5} (30)} \cdot \frac{\log_{2} (30) \log_{3} (30)}{\log_{2} (30) \log_{3} (30)}$

Multiply $\frac{1}{\log_{5} (30)}$ by $\frac{\log_{2} (30) \log_{3} (30)}{\log_{2} (30) \log_{3} (30)}$ .

$\frac{(\log_{3} (30) + \log_{2} (30)) \log_{5} (30)}{\log_{2} (30) \log_{3} (30) \log_{5} (30)} + \frac{\log_{2} (30) \log_{3} (30)}{\log_{5} (30) (\log_{2} (30) \log_{3} (30))}$

Reorder the factors of $\log_{5} (30) (\log_{2} (30) \log_{3} (30))$ .

$\frac{(\log_{3} (30) + \log_{2} (30)) \log_{5} (30)}{\log_{2} (30) \log_{3} (30) \log_{5} (30)} + \frac{\log_{2} (30) \log_{3} (30)}{\log_{2} (30) \log_{3} (30) \log_{5} (30)}$

Step 8

Combine the numerators over the common denominator.

$\frac{(\log_{3} (30) + \log_{2} (30)) \log_{5} (30) + \log_{2} (30) \log_{3} (30)}{\log_{2} (30) \log_{3} (30) \log_{5} (30)}$

Step 9

Apply the distributive property.

$\frac{\log_{3} (30) \log_{5} (30) + \log_{2} (30) \log_{5} (30) + \log_{2} (30) \log_{3} (30)}{\log_{2} (30) \log_{3} (30) \log_{5} (30)}$

Step 10

The result can be shown in multiple forms.

Exact Form:

$\frac{\log_{3} (30) \log_{5} (30) + \log_{2} (30) \log_{5} (30) + \log_{2} (30) \log_{3} (30)}{\log_{2} (30) \log_{3} (30) \log_{5} (30)}$

Decimal Form:

$1$

Find the Exact Value 1/( log base 2 of 30)+1/( log base 3 of 30)+1/( log base 5 of 30)

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