Finite Math Examples

${(\frac{3}{4})}^{x} = \frac{64}{27}$

Step 1

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

$\ln ({(\frac{3}{4})}^{x}) = \ln (\frac{64}{27})$

Step 2

Expand the left side.

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Step 2.1

Expand $\ln ({(\frac{3}{4})}^{x})$ by moving $x$ outside the logarithm.

$x \ln (\frac{3}{4}) = \ln (\frac{64}{27})$

Step 2.2

Rewrite $\ln (\frac{3}{4})$ as $\ln (3) - \ln (4)$ .

$x (\ln (3) - \ln (4)) = \ln (\frac{64}{27})$

Step 2.3

Rewrite $\ln (4)$ as $\ln (2^{2})$ .

$x (\ln (3) - \ln (2^{2})) = \ln (\frac{64}{27})$

Step 2.4

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

$x (\ln (3) - (2 \ln (2))) = \ln (\frac{64}{27})$

Step 2.5

Multiply $2$ by $- 1$ .

$x (\ln (3) - 2 \ln (2)) = \ln (\frac{64}{27})$

Step 3

Simplify the left side.

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Step 3.1

Simplify $x (\ln (3) - 2 \ln (2))$ .

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Step 3.1.1

Apply the distributive property.

$x \ln (3) + x (- 2 \ln (2)) = \ln (\frac{64}{27})$

Step 3.1.2

Reorder factors in $x \ln (3) + x (- 2 \ln (2))$ .

$x \ln (3) - 2 x \ln (2) = \ln (\frac{64}{27})$

Step 4

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

$x \ln (3) - 2 x \ln (2) - \ln (\frac{64}{27}) = 0$

Step 5

Add $\ln (\frac{64}{27})$ to both sides of the equation.

$x \ln (3) - 2 x \ln (2) = \ln (\frac{64}{27})$

Step 6

Factor $x$ out of $x \ln (3) - 2 x \ln (2)$ .

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Step 6.1

Factor $x$ out of $x \ln (3)$ .

$x (\ln (3)) - 2 x \ln (2) = \ln (\frac{64}{27})$

Step 6.2

Factor $x$ out of $- 2 x \ln (2)$ .

$x (\ln (3)) + x (- 2 \ln (2)) = \ln (\frac{64}{27})$

Step 6.3

Factor $x$ out of $x (\ln (3)) + x (- 2 \ln (2))$ .

$x (\ln (3) - 2 \ln (2)) = \ln (\frac{64}{27})$

Step 7

Divide each term in $x (\ln (3) - 2 \ln (2)) = \ln (\frac{64}{27})$ by $\ln (3) - 2 \ln (2)$ and simplify.

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Step 7.1

Divide each term in $x (\ln (3) - 2 \ln (2)) = \ln (\frac{64}{27})$ by $\ln (3) - 2 \ln (2)$ .

$\frac{x (\ln (3) - 2 \ln (2))}{\ln (3) - 2 \ln (2)} = \frac{\ln (\frac{64}{27})}{\ln (3) - 2 \ln (2)}$

Step 7.2

Simplify the left side.

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Step 7.2.1

Cancel the common factor of $\ln (3) - 2 \ln (2)$ .

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Step 7.2.1.1

Cancel the common factor.

$\frac{x (\ln (3) - 2 \ln (2))}{\ln (3) - 2 \ln (2)} = \frac{\ln (\frac{64}{27})}{\ln (3) - 2 \ln (2)}$

Step 7.2.1.2

Divide $x$ by $1$ .

$x = \frac{\ln (\frac{64}{27})}{\ln (3) - 2 \ln (2)}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$x = \frac{\ln (\frac{64}{27})}{\ln (3) - 2 \ln (2)}$

Decimal Form:

$x = - 3$