Precalculus Examples

$- \frac{7}{8} \cdot \log_{6} (6 m^{2}) + \frac{1}{2} \cdot \log_{6} (36 m^{2})$

Step 1

Simplify each term.

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Step 1.1

Multiply $- \frac{7}{8} \log_{6} (6 m^{2})$ .

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Step 1.1.1

Reorder $\log_{6} (6 m^{2})$ and $\frac{7}{8}$ .

$- (\frac{7}{8} \log_{6} (6 m^{2})) + \frac{1}{2} \cdot \log_{6} (36 m^{2})$

Step 1.1.2

Simplify $\frac{7}{8} \log_{6} (6 m^{2})$ by moving $\frac{7}{8}$ inside the logarithm.

$- \log_{6} ({(6 m^{2})}^{\frac{7}{8}}) + \frac{1}{2} \cdot \log_{6} (36 m^{2})$

Step 1.2

Apply the product rule to $6 m^{2}$ .

$- \log_{6} (6^{\frac{7}{8}} {(m^{2})}^{\frac{7}{8}}) + \frac{1}{2} \cdot \log_{6} (36 m^{2})$

Step 1.3

Rewrite $\log_{6} (6^{\frac{7}{8}} {(m^{2})}^{\frac{7}{8}})$ as $\log_{6} (6^{\frac{7}{8}}) + \log_{6} ({(m^{2})}^{\frac{7}{8}})$ .

$- (\log_{6} (6^{\frac{7}{8}}) + \log_{6} ({(m^{2})}^{\frac{7}{8}})) + \frac{1}{2} \cdot \log_{6} (36 m^{2})$

Step 1.4

Use logarithm rules to move $\frac{7}{8}$ out of the exponent.

$- (\frac{7}{8} \log_{6} (6) + \log_{6} ({(m^{2})}^{\frac{7}{8}})) + \frac{1}{2} \cdot \log_{6} (36 m^{2})$

Step 1.5

Logarithm base $6$ of $6$ is $1$ .

$- (\frac{7}{8} \cdot 1 + \log_{6} ({(m^{2})}^{\frac{7}{8}})) + \frac{1}{2} \cdot \log_{6} (36 m^{2})$

Step 1.6

Multiply $\frac{7}{8}$ by $1$ .

$- (\frac{7}{8} + \log_{6} ({(m^{2})}^{\frac{7}{8}})) + \frac{1}{2} \cdot \log_{6} (36 m^{2})$

Step 1.7

Multiply the exponents in ${(m^{2})}^{\frac{7}{8}}$ .

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Step 1.7.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$- (\frac{7}{8} + \log_{6} (m^{2 (\frac{7}{8})})) + \frac{1}{2} \cdot \log_{6} (36 m^{2})$

Step 1.7.2

Cancel the common factor of $2$ .

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Step 1.7.2.1

Factor $2$ out of $8$ .

$- (\frac{7}{8} + \log_{6} (m^{2 \frac{7}{2 (4)}})) + \frac{1}{2} \cdot \log_{6} (36 m^{2})$

Step 1.7.2.2

Cancel the common factor.

$- (\frac{7}{8} + \log_{6} (m^{2 \frac{7}{2 \cdot 4}})) + \frac{1}{2} \cdot \log_{6} (36 m^{2})$

Step 1.7.2.3

Rewrite the expression.

$- (\frac{7}{8} + \log_{6} (m^{\frac{7}{4}})) + \frac{1}{2} \cdot \log_{6} (36 m^{2})$

Step 1.8

Apply the distributive property.

$- \frac{7}{8} - \log_{6} (m^{\frac{7}{4}}) + \frac{1}{2} \cdot \log_{6} (36 m^{2})$

Step 1.9

Simplify $\frac{1}{2} \log_{6} (36 m^{2})$ by moving $\frac{1}{2}$ inside the logarithm.

$- \frac{7}{8} - \log_{6} (m^{\frac{7}{4}}) + \log_{6} ({(36 m^{2})}^{\frac{1}{2}})$

Step 1.10

Apply the product rule to $36 m^{2}$ .

$- \frac{7}{8} - \log_{6} (m^{\frac{7}{4}}) + \log_{6} (36^{\frac{1}{2}} {(m^{2})}^{\frac{1}{2}})$

Step 1.11

Rewrite $36$ as $6^{2}$ .

$- \frac{7}{8} - \log_{6} (m^{\frac{7}{4}}) + \log_{6} ({(6^{2})}^{\frac{1}{2}} {(m^{2})}^{\frac{1}{2}})$

Step 1.12

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$- \frac{7}{8} - \log_{6} (m^{\frac{7}{4}}) + \log_{6} (6^{2 (\frac{1}{2})} {(m^{2})}^{\frac{1}{2}})$

Step 1.13

Cancel the common factor of $2$ .

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Step 1.13.1

Cancel the common factor.

$- \frac{7}{8} - \log_{6} (m^{\frac{7}{4}}) + \log_{6} (6^{2 (\frac{1}{2})} {(m^{2})}^{\frac{1}{2}})$

Step 1.13.2

Rewrite the expression.

$- \frac{7}{8} - \log_{6} (m^{\frac{7}{4}}) + \log_{6} (6^{1} {(m^{2})}^{\frac{1}{2}})$

Step 1.14

Evaluate the exponent.

$- \frac{7}{8} - \log_{6} (m^{\frac{7}{4}}) + \log_{6} (6 {(m^{2})}^{\frac{1}{2}})$

Step 1.15

Multiply the exponents in ${(m^{2})}^{\frac{1}{2}}$ .

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Step 1.15.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$- \frac{7}{8} - \log_{6} (m^{\frac{7}{4}}) + \log_{6} (6 m^{2 (\frac{1}{2})})$

Step 1.15.2

Cancel the common factor of $2$ .

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Step 1.15.2.1

Cancel the common factor.

$- \frac{7}{8} - \log_{6} (m^{\frac{7}{4}}) + \log_{6} (6 m^{2 (\frac{1}{2})})$

Step 1.15.2.2

Rewrite the expression.

$- \frac{7}{8} - \log_{6} (m^{\frac{7}{4}}) + \log_{6} (6 m^{1})$

Step 1.16

Simplify.

$- \frac{7}{8} - \log_{6} (m^{\frac{7}{4}}) + \log_{6} (6 m)$

Step 2

Simplify each term.

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Step 2.1

Rewrite $\log_{6} (m^{\frac{7}{4}})$ using the change of base formula.

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Step 2.1.1

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 $0$ .

$- \frac{7}{8} - \log_{a} (x) = \frac{\log_{b} (x)}{\log_{b} (a)} + \log_{6} (6 m)$

Step 2.1.2

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

$- \frac{7}{8} - \frac{\log (m^{\frac{7}{4}})}{\log (6)} + \log_{6} (6 m)$

Step 2.2

Rewrite $\log_{6} (6 m)$ using the change of base formula.

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Step 2.2.1

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 $0$ .

$- \frac{7}{8} - \frac{\log (m^{\frac{7}{4}})}{\log (6)} + \log_{a} (x) = \frac{\log_{b} (x)}{\log_{b} (a)}$

Step 2.2.2

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

$- \frac{7}{8} - \frac{\log (m^{\frac{7}{4}})}{\log (6)} + \frac{\log (6 m)}{\log (6)}$