Algebra Examples

$16^{- 3 y} \cdot \frac{1}{8} = {(\frac{1}{32})}^{2 y - 3}$

Step 1

Take the log of both sides of the equation.

$\ln (16^{- 3 y} \cdot \frac{1}{8}) = \ln ({(\frac{1}{32})}^{2 y - 3})$

Step 2

Rewrite $\ln (16^{- 3 y} \cdot \frac{1}{8})$ as $\ln (16^{- 3 y}) + \ln (\frac{1}{8})$ .

$\ln (16^{- 3 y}) + \ln (\frac{1}{8}) = \ln ({(\frac{1}{32})}^{2 y - 3})$

Step 3

Rewrite $\ln (\frac{1}{8})$ as $\ln (1) - \ln (8)$ .

$\ln (16^{- 3 y}) + \ln (1) - \ln (8) = \ln ({(\frac{1}{32})}^{2 y - 3})$

Step 4

Expand $\ln (16^{- 3 y})$ by moving $- 3 y$ outside the logarithm.

$- 3 y \ln (16) + \ln (1) - \ln (8) = \ln ({(\frac{1}{32})}^{2 y - 3})$

Step 5

The natural logarithm of $1$ is $0$ .

$- 3 y \ln (16) + 0 - \ln (8) = \ln ({(\frac{1}{32})}^{2 y - 3})$

Step 6

Rewrite $\ln (8)$ as $\ln (2^{3})$ .

$- 3 y \ln (16) + 0 - \ln (2^{3}) = \ln ({(\frac{1}{32})}^{2 y - 3})$

Step 7

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

$- 3 y \ln (16) + 0 - (3 \ln (2)) = \ln ({(\frac{1}{32})}^{2 y - 3})$

Step 8

Multiply $3$ by $- 1$ .

$- 3 y \ln (16) + 0 - 3 \ln (2) = \ln ({(\frac{1}{32})}^{2 y - 3})$

Step 9

Subtract $3 \ln (2)$ from $0$ .

$- 3 y \ln (16) - 3 \ln (2) = \ln ({(\frac{1}{32})}^{2 y - 3})$

Step 10

Expand $\ln ({(\frac{1}{32})}^{2 y - 3})$ by moving $2 y - 3$ outside the logarithm.

$- 3 y \ln (16) - 3 \ln (2) = (2 y - 3) \ln (\frac{1}{32})$

Step 11

Rewrite $\ln (\frac{1}{32})$ as $\ln (1) - \ln (32)$ .

$- 3 y \ln (16) - 3 \ln (2) = (2 y - 3) (\ln (1) - \ln (32))$

Step 12

The natural logarithm of $1$ is $0$ .

$- 3 y \ln (16) - 3 \ln (2) = (2 y - 3) (0 - \ln (32))$

Step 13

Rewrite $\ln (32)$ as $\ln (2^{5})$ .

$- 3 y \ln (16) - 3 \ln (2) = (2 y - 3) (0 - \ln (2^{5}))$

Step 14

Expand $\ln (2^{5})$ by moving $5$ outside the logarithm.

$- 3 y \ln (16) - 3 \ln (2) = (2 y - 3) (0 - (5 \ln (2)))$

Step 15

Multiply $5$ by $- 1$ .

$- 3 y \ln (16) - 3 \ln (2) = (2 y - 3) (0 - 5 \ln (2))$

Step 16

Subtract $5 \ln (2)$ from $0$ .

$- 3 y \ln (16) - 3 \ln (2) = (2 y - 3) (- 5 \ln (2))$

Step 17

Solve the equation for $y$ .

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Step 17.1

Simplify the right side.

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Step 17.1.1

Apply the distributive property.

$- 3 y \ln (16) - 3 \ln (2) = 2 y (- 5 \ln (2)) - 3 (- 5 \ln (2))$

Step 17.1.2

Multiply.

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Step 17.1.2.1

Multiply $- 5$ by $2$ .

$- 3 y \ln (16) - 3 \ln (2) = - 10 y \ln (2) - 3 (- 5 \ln (2))$

Step 17.1.2.2

Multiply $- 5$ by $- 3$ .

$- 3 y \ln (16) - 3 \ln (2) = - 10 y \ln (2) + 15 \ln (2)$

Step 17.2

Simplify the left side.

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Step 17.2.1

Simplify $- 3 y \ln (16) - 3 \ln (2)$ .

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Step 17.2.1.1

Simplify each term.

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Step 17.2.1.1.1

Simplify $- 3 \ln (16)$ by moving $3$ inside the logarithm.

$y (- \ln (16^{3})) - 3 \ln (2) = - 10 y \ln (2) + 15 \ln (2)$

Step 17.2.1.1.2

Raise $16$ to the power of $3$ .

$y (- \ln (4096)) - 3 \ln (2) = - 10 y \ln (2) + 15 \ln (2)$

Step 17.2.1.1.3

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

$y (- \ln (4096)) - \ln (2^{3}) = - 10 y \ln (2) + 15 \ln (2)$

Step 17.2.1.1.4

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

$y (- \ln (4096)) - \ln (8) = - 10 y \ln (2) + 15 \ln (2)$

Step 17.2.1.2

Reorder factors in $y (- \ln (4096)) - \ln (8)$ .

$- y \ln (4096) - \ln (8) = - 10 y \ln (2) + 15 \ln (2)$

Step 17.3

Simplify the right side.

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Step 17.3.1

Simplify $- 10 y \ln (2) + 15 \ln (2)$ .

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Step 17.3.1.1

Simplify each term.

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Step 17.3.1.1.1

Simplify $- 10 \ln (2)$ by moving $10$ inside the logarithm.

$- y \ln (4096) - \ln (8) = y (- \ln (2^{10})) + 15 \ln (2)$

Step 17.3.1.1.2

Raise $2$ to the power of $10$ .

$- y \ln (4096) - \ln (8) = y (- \ln (1024)) + 15 \ln (2)$

Step 17.3.1.1.3

Simplify $15 \ln (2)$ by moving $15$ inside the logarithm.

$- y \ln (4096) - \ln (8) = y (- \ln (1024)) + \ln (2^{15})$

Step 17.3.1.1.4

Raise $2$ to the power of $15$ .

$- y \ln (4096) - \ln (8) = y (- \ln (1024)) + \ln (32768)$

Step 17.3.1.2

Reorder factors in $y (- \ln (1024)) + \ln (32768)$ .

$- y \ln (4096) - \ln (8) = - y \ln (1024) + \ln (32768)$

Step 17.4

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

$- y \ln (4096) - \ln (8) + y \ln (1024) - \ln (32768) = 0$

Step 17.5

Move all terms not containing $y$ to the right side of the equation.

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Step 17.5.1

Add $\ln (8)$ to both sides of the equation.

$- y \ln (4096) + y \ln (1024) - \ln (32768) = \ln (8)$

Step 17.5.2

Add $\ln (32768)$ to both sides of the equation.

$- y \ln (4096) + y \ln (1024) = \ln (8) + \ln (32768)$

Step 17.6

Factor $y$ out of $- y \ln (4096) + y \ln (1024)$ .

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Step 17.6.1

Factor $y$ out of $- y \ln (4096)$ .

$y (- 1 \ln (4096)) + y \ln (1024) = \ln (8) + \ln (32768)$

Step 17.6.2

Factor $y$ out of $y \ln (1024)$ .

$y (- 1 \ln (4096)) + y (\ln (1024)) = \ln (8) + \ln (32768)$

Step 17.6.3

Factor $y$ out of $y (- 1 \ln (4096)) + y (\ln (1024))$ .

$y (- 1 \ln (4096) + \ln (1024)) = \ln (8) + \ln (32768)$

Step 17.7

Rewrite $- 1 \ln (4096)$ as $- \ln (4096)$ .

$y (- \ln (4096) + \ln (1024)) = \ln (8) + \ln (32768)$

Step 17.8

Divide each term in $y (- \ln (4096) + \ln (1024)) = \ln (8) + \ln (32768)$ by $- \ln (4096) + \ln (1024)$ and simplify.

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Step 17.8.1

Divide each term in $y (- \ln (4096) + \ln (1024)) = \ln (8) + \ln (32768)$ by $- \ln (4096) + \ln (1024)$ .

$\frac{y (- \ln (4096) + \ln (1024))}{- \ln (4096) + \ln (1024)} = \frac{\ln (8)}{- \ln (4096) + \ln (1024)} + \frac{\ln (32768)}{- \ln (4096) + \ln (1024)}$

Step 17.8.2

Simplify the left side.

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Step 17.8.2.1

Cancel the common factor of $- \ln (4096) + \ln (1024)$ .

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Step 17.8.2.1.1

Cancel the common factor.

$\frac{y (- \ln (4096) + \ln (1024))}{- \ln (4096) + \ln (1024)} = \frac{\ln (8)}{- \ln (4096) + \ln (1024)} + \frac{\ln (32768)}{- \ln (4096) + \ln (1024)}$

Step 17.8.2.1.2

Divide $y$ by $1$ .

$y = \frac{\ln (8)}{- \ln (4096) + \ln (1024)} + \frac{\ln (32768)}{- \ln (4096) + \ln (1024)}$

Step 17.8.3

Simplify the right side.

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Step 17.8.3.1

Combine the numerators over the common denominator.

$y = \frac{\ln (8) + \ln (32768)}{- \ln (4096) + \ln (1024)}$

Step 17.8.3.2

Factor $- 1$ out of $- \ln (4096)$ .

$y = \frac{\ln (8) + \ln (32768)}{- \ln (4096) + \ln (1024)}$

Step 17.8.3.3

Factor $- 1$ out of $\ln (1024)$ .

$y = \frac{\ln (8) + \ln (32768)}{- \ln (4096) - 1 (- \ln (1024))}$

Step 17.8.3.4

Factor $- 1$ out of $- \ln (4096) - 1 (- \ln (1024))$ .

$y = \frac{\ln (8) + \ln (32768)}{- (\ln (4096) - \ln (1024))}$

Step 17.8.3.5

Rewrite negatives.

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Step 17.8.3.5.1

Rewrite $- (\ln (4096) - \ln (1024))$ as $- 1 (\ln (4096) - \ln (1024))$ .

$y = \frac{\ln (8) + \ln (32768)}{- 1 (\ln (4096) - \ln (1024))}$

Step 17.8.3.5.2

Move the negative in front of the fraction.

$y = - \frac{\ln (8) + \ln (32768)}{\ln (4096) - \ln (1024)}$

Step 18

The result can be shown in multiple forms.

Exact Form:

$y = - \frac{\ln (8) + \ln (32768)}{\ln (4096) - \ln (1024)}$

Decimal Form:

$y = - 9.000000000000 \dots$