Basic Math Examples

$6^{a + 2} = 64 \cdot 3^{a + 2}$

Step 1

Take the log of both sides of the equation.

$\ln (6^{a + 2}) = \ln (64 \cdot 3^{a + 2})$

Step 2

Expand $\ln (6^{a + 2})$ by moving $a + 2$ outside the logarithm.

$(a + 2) \ln (6) = \ln (64 \cdot 3^{a + 2})$

Step 3

Rewrite $\ln (64 \cdot 3^{a + 2})$ as $\ln (64) + \ln (3^{a + 2})$ .

$(a + 2) \ln (6) = \ln (64) + \ln (3^{a + 2})$

Step 4

Expand $\ln (3^{a + 2})$ by moving $a + 2$ outside the logarithm.

$(a + 2) \ln (6) = \ln (64) + (a + 2) \ln (3)$

Step 5

Solve the equation for $a$ .

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Step 5.1

Simplify the left side.

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Step 5.1.1

Simplify $(a + 2) \ln (6)$ .

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Step 5.1.1.1

Apply the distributive property.

$a \ln (6) + 2 \ln (6) = \ln (64) + (a + 2) \ln (3)$

Step 5.1.1.2

Simplify $2 \ln (6)$ by moving $2$ inside the logarithm.

$a \ln (6) + \ln (6^{2}) = \ln (64) + (a + 2) \ln (3)$

Step 5.1.1.3

Raise $6$ to the power of $2$ .

$a \ln (6) + \ln (36) = \ln (64) + (a + 2) \ln (3)$

Step 5.2

Simplify the right side.

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Step 5.2.1

Simplify $\ln (64) + (a + 2) \ln (3)$ .

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Step 5.2.1.1

Simplify each term.

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Step 5.2.1.1.1

Apply the distributive property.

$a \ln (6) + \ln (36) = \ln (64) + a \ln (3) + 2 \ln (3)$

Step 5.2.1.1.2

Simplify $2 \ln (3)$ by moving $2$ inside the logarithm.

$a \ln (6) + \ln (36) = \ln (64) + a \ln (3) + \ln (3^{2})$

Step 5.2.1.1.3

Raise $3$ to the power of $2$ .

$a \ln (6) + \ln (36) = \ln (64) + a \ln (3) + \ln (9)$

Step 5.2.1.2

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

$a \ln (6) + \ln (36) = a \ln (3) + \ln (64 \cdot 9)$

Step 5.2.1.3

Multiply $64$ by $9$ .

$a \ln (6) + \ln (36) = a \ln (3) + \ln (576)$

Step 5.3

Move all the terms containing a logarithm to the left side of the equation.

$a \ln (6) + \ln (36) - a \ln (3) - \ln (576) = 0$

Step 5.4

Use the quotient property of logarithms, $\log_{b} (x) - \log_{b} (y) = \log_{b} (\frac{x}{y})$ .

$a \ln (6) - a \ln (3) + \ln (\frac{36}{576}) = 0$

Step 5.5

Cancel the common factor of $36$ and $576$ .

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Step 5.5.1

Factor $36$ out of $36$ .

$a \ln (6) - a \ln (3) + \ln (\frac{36 (1)}{576}) = 0$

Step 5.5.2

Cancel the common factors.

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Step 5.5.2.1

Factor $36$ out of $576$ .

$a \ln (6) - a \ln (3) + \ln (\frac{36 \cdot 1}{36 \cdot 16}) = 0$

Step 5.5.2.2

Cancel the common factor.

$a \ln (6) - a \ln (3) + \ln (\frac{36 \cdot 1}{36 \cdot 16}) = 0$

Step 5.5.2.3

Rewrite the expression.

$a \ln (6) - a \ln (3) + \ln (\frac{1}{16}) = 0$

Step 5.6

Subtract $\ln (\frac{1}{16})$ from both sides of the equation.

$a \ln (6) - a \ln (3) = - \ln (\frac{1}{16})$

Step 5.7

Factor $a$ out of $a \ln (6) - a \ln (3)$ .

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Step 5.7.1

Factor $a$ out of $a \ln (6)$ .

$a (\ln (6)) - a \ln (3) = - \ln (\frac{1}{16})$

Step 5.7.2

Factor $a$ out of $- a \ln (3)$ .

$a (\ln (6)) + a (- 1 \ln (3)) = - \ln (\frac{1}{16})$

Step 5.7.3

Factor $a$ out of $a (\ln (6)) + a (- 1 \ln (3))$ .

$a (\ln (6) - 1 \ln (3)) = - \ln (\frac{1}{16})$

Step 5.8

Rewrite $- 1 \ln (3)$ as $- \ln (3)$ .

$a (\ln (6) - \ln (3)) = - \ln (\frac{1}{16})$

Step 5.9

Divide each term in $a (\ln (6) - \ln (3)) = - \ln (\frac{1}{16})$ by $\ln (6) - \ln (3)$ and simplify.

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Step 5.9.1

Divide each term in $a (\ln (6) - \ln (3)) = - \ln (\frac{1}{16})$ by $\ln (6) - \ln (3)$ .

$\frac{a (\ln (6) - \ln (3))}{\ln (6) - \ln (3)} = \frac{- \ln (\frac{1}{16})}{\ln (6) - \ln (3)}$

Step 5.9.2

Simplify the left side.

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Step 5.9.2.1

Cancel the common factor of $\ln (6) - \ln (3)$ .

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Step 5.9.2.1.1

Cancel the common factor.

$\frac{a (\ln (6) - \ln (3))}{\ln (6) - \ln (3)} = \frac{- \ln (\frac{1}{16})}{\ln (6) - \ln (3)}$

Step 5.9.2.1.2

Divide $a$ by $1$ .

$a = \frac{- \ln (\frac{1}{16})}{\ln (6) - \ln (3)}$

Step 5.9.3

Simplify the right side.

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Step 5.9.3.1

Move the negative in front of the fraction.

$a = - \frac{\ln (\frac{1}{16})}{\ln (6) - \ln (3)}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$a = - \frac{\ln (\frac{1}{16})}{\ln (6) - \ln (3)}$

Decimal Form:

$a = 4$