Precalculus Examples

$\frac{1}{\log_{5} (35)} + \frac{1}{\log_{7} (35)}$

Step 1

To write $\frac{1}{\log_{5} (35)}$ as a fraction with a common denominator, multiply by $\frac{\log_{7} (35)}{\log_{7} (35)}$ .

$\frac{1}{\log_{5} (35)} \cdot \frac{\log_{7} (35)}{\log_{7} (35)} + \frac{1}{\log_{7} (35)}$

Step 2

To write $\frac{1}{\log_{7} (35)}$ as a fraction with a common denominator, multiply by $\frac{\log_{5} (35)}{\log_{5} (35)}$ .

$\frac{1}{\log_{5} (35)} \cdot \frac{\log_{7} (35)}{\log_{7} (35)} + \frac{1}{\log_{7} (35)} \cdot \frac{\log_{5} (35)}{\log_{5} (35)}$

Step 3

Write each expression with a common denominator of $\log_{5} (35) \log_{7} (35)$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.1

Multiply $\frac{1}{\log_{5} (35)}$ by $\frac{\log_{7} (35)}{\log_{7} (35)}$ .

$\frac{\log_{7} (35)}{\log_{5} (35) \log_{7} (35)} + \frac{1}{\log_{7} (35)} \cdot \frac{\log_{5} (35)}{\log_{5} (35)}$

Step 3.2

Multiply $\frac{1}{\log_{7} (35)}$ by $\frac{\log_{5} (35)}{\log_{5} (35)}$ .

$\frac{\log_{7} (35)}{\log_{5} (35) \log_{7} (35)} + \frac{\log_{5} (35)}{\log_{7} (35) \log_{5} (35)}$

Step 3.3

Reorder the factors of $\log_{7} (35) \log_{5} (35)$ .

$\frac{\log_{7} (35)}{\log_{5} (35) \log_{7} (35)} + \frac{\log_{5} (35)}{\log_{5} (35) \log_{7} (35)}$

Step 4

Combine the numerators over the common denominator.

$\frac{\log_{7} (35) + \log_{5} (35)}{\log_{5} (35) \log_{7} (35)}$

Step 5

Simplify the numerator.

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Step 5.1

Rewrite $\log_{7} (35)$ using the change of base formula.

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Step 5.1.1

The change of base rule can be used if $a$ and $b$ are greater than $0$ and not equal to $1$ , and $x$ is greater than $0$ .

$\frac{\log_{a} (x) = \frac{\log_{b} (x)}{\log_{b} (a)} + \log_{5} (35)}{\log_{5} (35) \log_{7} (35)}$

Step 5.1.2

Substitute in values for the variables in the change of base formula, using $b = 10$ .

$\frac{\frac{\log (35)}{\log (7)} + \log_{5} (35)}{\log_{5} (35) \log_{7} (35)}$

Step 5.2

Rewrite $\log_{5} (35)$ using the change of base formula.

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Step 5.2.1

The change of base rule can be used if $a$ and $b$ are greater than $0$ and not equal to $1$ , and $x$ is greater than $0$ .

$\frac{\frac{\log (35)}{\log (7)} + \log_{a} (x) = \frac{\log_{b} (x)}{\log_{b} (a)}}{\log_{5} (35) \log_{7} (35)}$

Step 5.2.2

Substitute in values for the variables in the change of base formula, using $b = 10$ .

$\frac{\frac{\log (35)}{\log (7)} + \frac{\log (35)}{\log (5)}}{\log_{5} (35) \log_{7} (35)}$

Step 5.3

To write $\frac{\log (35)}{\log (7)}$ as a fraction with a common denominator, multiply by $\frac{\log (5)}{\log (5)}$ .

$\frac{\frac{\log (35)}{\log (7)} \cdot \frac{\log (5)}{\log (5)} + \frac{\log (35)}{\log (5)}}{\log_{5} (35) \log_{7} (35)}$

Step 5.4

To write $\frac{\log (35)}{\log (5)}$ as a fraction with a common denominator, multiply by $\frac{\log (7)}{\log (7)}$ .

$\frac{\frac{\log (35)}{\log (7)} \cdot \frac{\log (5)}{\log (5)} + \frac{\log (35)}{\log (5)} \cdot \frac{\log (7)}{\log (7)}}{\log_{5} (35) \log_{7} (35)}$

Step 5.5

Write each expression with a common denominator of $\log (7) \log (5)$ , by multiplying each by an appropriate factor of $1$ .

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Step 5.5.1

Multiply $\frac{\log (35)}{\log (7)}$ by $\frac{\log (5)}{\log (5)}$ .

$\frac{\frac{\log (35) \log (5)}{\log (7) \log (5)} + \frac{\log (35)}{\log (5)} \cdot \frac{\log (7)}{\log (7)}}{\log_{5} (35) \log_{7} (35)}$

Step 5.5.2

Multiply $\frac{\log (35)}{\log (5)}$ by $\frac{\log (7)}{\log (7)}$ .

$\frac{\frac{\log (35) \log (5)}{\log (7) \log (5)} + \frac{\log (35) \log (7)}{\log (5) \log (7)}}{\log_{5} (35) \log_{7} (35)}$

Step 5.5.3

Reorder the factors of $\log (7) \log (5)$ .

$\frac{\frac{\log (35) \log (5)}{\log (5) \log (7)} + \frac{\log (35) \log (7)}{\log (5) \log (7)}}{\log_{5} (35) \log_{7} (35)}$

Step 5.6

Combine the numerators over the common denominator.

$\frac{\frac{\log (35) \log (5) + \log (35) \log (7)}{\log (5) \log (7)}}{\log_{5} (35) \log_{7} (35)}$

Step 6

Simplify the denominator.

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Step 6.1

Rewrite $\log_{5} (35)$ using the change of base formula.

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Step 6.1.1

The change of base rule can be used if $a$ and $b$ are greater than $0$ and not equal to $1$ , and $x$ is greater than $0$ .

$\frac{\frac{\log (35) \log (5) + \log (35) \log (7)}{\log (5) \log (7)}}{\log_{a} (x) = \frac{\log_{b} (x)}{\log_{b} (a)} \log_{7} (35)}$

Step 6.1.2

Substitute in values for the variables in the change of base formula, using $b = 10$ .

$\frac{\frac{\log (35) \log (5) + \log (35) \log (7)}{\log (5) \log (7)}}{\frac{\log (35)}{\log (5)} \log_{7} (35)}$

Step 6.2

Rewrite $\log_{7} (35)$ using the change of base formula.

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Step 6.2.1

The change of base rule can be used if $a$ and $b$ are greater than $0$ and not equal to $1$ , and $x$ is greater than $0$ .

$\frac{\frac{\log (35) \log (5) + \log (35) \log (7)}{\log (5) \log (7)}}{\frac{\log (35)}{\log (5)} \log_{a} (x) = \frac{\log_{b} (x)}{\log_{b} (a)}}$

Step 6.2.2

Substitute in values for the variables in the change of base formula, using $b = 10$ .

$\frac{\frac{\log (35) \log (5) + \log (35) \log (7)}{\log (5) \log (7)}}{\frac{\log (35)}{\log (5)} \cdot \frac{\log (35)}{\log (7)}}$

Step 7

Multiply $\frac{\log (35)}{\log (5)}$ by $\frac{\log (35)}{\log (7)}$ .

$\frac{\frac{\log (35) \log (5) + \log (35) \log (7)}{\log (5) \log (7)}}{\frac{\log (35) \log (35)}{\log (5) \log (7)}}$

Step 8

Simplify the numerator.

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Step 8.1

Raise $\log (35)$ to the power of $1$ .

$\frac{\frac{\log (35) \log (5) + \log (35) \log (7)}{\log (5) \log (7)}}{\frac{\log^{1} (35) \log (35)}{\log (5) \log (7)}}$

Step 8.2

Raise $\log (35)$ to the power of $1$ .

$\frac{\frac{\log (35) \log (5) + \log (35) \log (7)}{\log (5) \log (7)}}{\frac{\log^{1} (35) \log^{1} (35)}{\log (5) \log (7)}}$

Step 8.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{\frac{\log (35) \log (5) + \log (35) \log (7)}{\log (5) \log (7)}}{\frac{{\log (35)}^{1 + 1}}{\log (5) \log (7)}}$

Step 8.4

Add $1$ and $1$ .

$\frac{\frac{\log (35) \log (5) + \log (35) \log (7)}{\log (5) \log (7)}}{\frac{\log^{2} (35)}{\log (5) \log (7)}}$

Step 9

Multiply the numerator by the reciprocal of the denominator.

$\frac{\log (35) \log (5) + \log (35) \log (7)}{\log (5) \log (7)} \cdot \frac{\log (5) \log (7)}{\log^{2} (35)}$

Step 10

Cancel the common factor of $\log (5) \log (7)$ .

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Step 10.1

Cancel the common factor.

$\frac{\log (35) \log (5) + \log (35) \log (7)}{\log (5) \log (7)} \cdot \frac{\log (5) \log (7)}{\log^{2} (35)}$

Step 10.2

Rewrite the expression.

$(\log (35) \log (5) + \log (35) \log (7)) \frac{1}{\log^{2} (35)}$

Step 11

Apply the distributive property.

$\log (35) \log (5) \frac{1}{\log^{2} (35)} + \log (35) \log (7) \frac{1}{\log^{2} (35)}$

Step 12

Cancel the common factor of $\log (35)$ .

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Step 12.1

Factor $\log (35)$ out of $\log (35) \log (5)$ .

$\log (35) (\log (5)) \frac{1}{\log^{2} (35)} + \log (35) \log (7) \frac{1}{\log^{2} (35)}$

Step 12.2

Factor $\log (35)$ out of $\log^{2} (35)$ .

$\log (35) (\log (5)) \frac{1}{\log (35) \log (35)} + \log (35) \log (7) \frac{1}{\log^{2} (35)}$

Step 12.3

Cancel the common factor.

$\log (35) \log (5) \frac{1}{\log (35) \log (35)} + \log (35) \log (7) \frac{1}{\log^{2} (35)}$

Step 12.4

Rewrite the expression.

$\log (5) \frac{1}{\log (35)} + \log (35) \log (7) \frac{1}{\log^{2} (35)}$

Step 13

Combine $\log (5)$ and $\frac{1}{\log (35)}$ .

$\frac{\log (5)}{\log (35)} + \log (35) \log (7) \frac{1}{\log^{2} (35)}$

Step 14

Cancel the common factor of $\log (35)$ .

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Step 14.1

Factor $\log (35)$ out of $\log (35) \log (7)$ .

$\frac{\log (5)}{\log (35)} + \log (35) (\log (7)) \frac{1}{\log^{2} (35)}$

Step 14.2

Factor $\log (35)$ out of $\log^{2} (35)$ .

$\frac{\log (5)}{\log (35)} + \log (35) (\log (7)) \frac{1}{\log (35) \log (35)}$

Step 14.3

Cancel the common factor.

$\frac{\log (5)}{\log (35)} + \log (35) \log (7) \frac{1}{\log (35) \log (35)}$

Step 14.4

Rewrite the expression.

$\frac{\log (5)}{\log (35)} + \log (7) \frac{1}{\log (35)}$

Step 15

Combine $\log (7)$ and $\frac{1}{\log (35)}$ .

$\frac{\log (5)}{\log (35)} + \frac{\log (7)}{\log (35)}$

Step 16

Combine the numerators over the common denominator.

$\frac{\log (5) + \log (7)}{\log (35)}$

Step 17

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

$\frac{\log (5 \cdot 7)}{\log (35)}$

Step 18

Multiply $5$ by $7$ .

$\frac{\log (35)}{\log (35)}$

Step 19

Cancel the common factor of $\log (35)$ .

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Step 19.1

Cancel the common factor.

$\frac{\log (35)}{\log (35)}$

Step 19.2

Rewrite the expression.

$1$