Precalculus Examples

$\frac{6 + \log_{7} (35)}{3}$

Step 1

Simplify the numerator.

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Step 1.1

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

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Step 1.1.1

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 $0$ .

$\frac{6 + \log_{a} (x) = \frac{\log_{b} (x)}{\log_{b} (a)}}{3}$

Step 1.1.2

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

$\frac{6 + \frac{\log (35)}{\log (7)}}{3}$

Step 1.2

To write $6$ as a fraction with a common denominator, multiply by $\frac{\log (7)}{\log (7)}$ .

$\frac{\frac{6 \log (7)}{\log (7)} + \frac{\log (35)}{\log (7)}}{3}$

Step 1.3

Combine the numerators over the common denominator.

$\frac{\frac{6 \log (7) + \log (35)}{\log (7)}}{3}$

Step 1.4

Simplify the numerator.

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Step 1.4.1

Simplify $6 \log (7)$ by moving $6$ inside the logarithm.

$\frac{\frac{\log (7^{6}) + \log (35)}{\log (7)}}{3}$

Step 1.4.2

Raise $7$ to the power of $6$ .

$\frac{\frac{\log (117649) + \log (35)}{\log (7)}}{3}$

Step 1.4.3

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

$\frac{\frac{\log (117649 \cdot 35)}{\log (7)}}{3}$

Step 1.4.4

Multiply $117649$ by $35$ .

$\frac{\frac{\log (4117715)}{\log (7)}}{3}$

Step 2

Multiply the numerator by the reciprocal of the denominator.

$\frac{\log (4117715)}{\log (7)} \cdot \frac{1}{3}$

Step 3

Multiply $\frac{\log (4117715)}{\log (7)} \cdot \frac{1}{3}$ .

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Step 3.1

Multiply $\frac{\log (4117715)}{\log (7)}$ by $\frac{1}{3}$ .

$\frac{\log (4117715)}{\log (7) \cdot 3}$

Step 3.2

Reorder $\log (7)$ and $3$ .

$\frac{\log (4117715)}{3 \cdot \log (7)}$

Step 3.3

Simplify $3 \cdot \log (7)$ by moving $3$ inside the logarithm.

$\frac{\log (4117715)}{\log (7^{3})}$

Step 4

Raise $7$ to the power of $3$ .

$\frac{\log (4117715)}{\log (343)}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$\frac{\log (4117715)}{\log (343)}$

Decimal Form:

$2.60902915 \dots$