Finite Math Examples

${(4 a^{b})}^{c} = 1024 a^{6 b}$

Step 1

Take the natural logarithm of both sides of the equation to remove the variable from the exponent.

$\ln ({(4 a^{b})}^{c}) = \ln (1024 a^{6 b})$

Step 2

Expand the left side.

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Step 2.1

Expand $\ln ({(4 a^{b})}^{c})$ by moving $c$ outside the logarithm.

$c \ln (4 a^{b}) = \ln (1024 a^{6 b})$

Step 2.2

Rewrite $\ln (4 a^{b})$ as $\ln (4) + \ln (a^{b})$ .

$c (\ln (4) + \ln (a^{b})) = \ln (1024 a^{6 b})$

Step 2.3

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

$c (\ln (4) + b \ln (a)) = \ln (1024 a^{6 b})$

Step 3

Expand the right side.

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Step 3.1

Rewrite $\ln (1024 a^{6 b})$ as $\ln (1024) + \ln (a^{6 b})$ .

$c (\ln (4) + b \ln (a)) = \ln (1024) + \ln (a^{6 b})$

Step 3.2

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

$c (\ln (4) + b \ln (a)) = \ln (1024) + 6 b \ln (a)$

Step 4

Simplify the left side.

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Step 4.1

Apply the distributive property.

$c \ln (4) + c b \ln (a) = \ln (1024) + 6 b \ln (a)$

Step 5

Simplify the left side.

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Step 5.1

Reorder $c$ and $b$ .

$c \ln (4) + b c \ln (a) = \ln (1024) + 6 b \ln (a)$

Step 5.2

Reorder $c \ln (4)$ and $b c \ln (a)$ .

$b c \ln (a) + c \ln (4) = \ln (1024) + 6 b \ln (a)$

Step 6

Reorder $\ln (1024)$ and $6 b \ln (a)$ .

$b c \ln (a) + c \ln (4) = 6 b \ln (a) + \ln (1024)$

Step 7

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

$b c \ln (a) + c \ln (4) - 6 b \ln (a) - \ln (1024) = 0$

Step 8

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

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Step 8.1

Add $6 b \ln (a)$ to both sides of the equation.

$b c \ln (a) + c \ln (4) - \ln (1024) = 6 b \ln (a)$

Step 8.2

Add $\ln (1024)$ to both sides of the equation.

$b c \ln (a) + c \ln (4) = 6 b \ln (a) + \ln (1024)$

Step 9

Factor $c$ out of $b c \ln (a) + c \ln (4)$ .

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Step 9.1

Factor $c$ out of $b c \ln (a)$ .

$c (b \ln (a)) + c \ln (4) = 6 b \ln (a) + \ln (1024)$

Step 9.2

Factor $c$ out of $c \ln (4)$ .

$c (b \ln (a)) + c (\ln (4)) = 6 b \ln (a) + \ln (1024)$

Step 9.3

Factor $c$ out of $c (b \ln (a)) + c (\ln (4))$ .

$c (b \ln (a) + \ln (4)) = 6 b \ln (a) + \ln (1024)$

Step 10

Divide each term in $c (b \ln (a) + \ln (4)) = 6 b \ln (a) + \ln (1024)$ by $b \ln (a) + \ln (4)$ and simplify.

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Step 10.1

Divide each term in $c (b \ln (a) + \ln (4)) = 6 b \ln (a) + \ln (1024)$ by $b \ln (a) + \ln (4)$ .

$\frac{c (b \ln (a) + \ln (4))}{b \ln (a) + \ln (4)} = \frac{6 b \ln (a)}{b \ln (a) + \ln (4)} + \frac{\ln (1024)}{b \ln (a) + \ln (4)}$

Step 10.2

Simplify the left side.

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Step 10.2.1

Cancel the common factor of $b \ln (a) + \ln (4)$ .

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Step 10.2.1.1

Cancel the common factor.

$\frac{c (b \ln (a) + \ln (4))}{b \ln (a) + \ln (4)} = \frac{6 b \ln (a)}{b \ln (a) + \ln (4)} + \frac{\ln (1024)}{b \ln (a) + \ln (4)}$

Step 10.2.1.2

Divide $c$ by $1$ .

$c = \frac{6 b \ln (a)}{b \ln (a) + \ln (4)} + \frac{\ln (1024)}{b \ln (a) + \ln (4)}$

Step 10.3

Simplify the right side.

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Step 10.3.1

Combine the numerators over the common denominator.

$c = \frac{6 b \ln (a) + \ln (1024)}{b \ln (a) + \ln (4)}$