Precalculus Examples

$(3^{4}) (5^{3 x - 2}) = (2^{5 - 2 x}) (7^{3})$

Step 1

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

$\ln (3^{4} \cdot 5^{3 x - 2}) = \ln (2^{5 - 2 x} \cdot 7^{3})$

Step 2

Expand the left side.

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Step 2.1

Rewrite $\ln (3^{4} \cdot 5^{3 x - 2})$ as $\ln (3^{4}) + \ln (5^{3 x - 2})$ .

$\ln (3^{4}) + \ln (5^{3 x - 2}) = \ln (2^{5 - 2 x} \cdot 7^{3})$

Step 2.2

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

$4 \ln (3) + \ln (5^{3 x - 2}) = \ln (2^{5 - 2 x} \cdot 7^{3})$

Step 2.3

Expand $\ln (5^{3 x - 2})$ by moving $3 x - 2$ outside the logarithm.

$4 \ln (3) + (3 x - 2) \ln (5) = \ln (2^{5 - 2 x} \cdot 7^{3})$

Step 3

Expand the right side.

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Step 3.1

Rewrite $\ln (2^{5 - 2 x} \cdot 7^{3})$ as $\ln (2^{5 - 2 x}) + \ln (7^{3})$ .

$4 \ln (3) + (3 x - 2) \ln (5) = \ln (2^{5 - 2 x}) + \ln (7^{3})$

Step 3.2

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

$4 \ln (3) + (3 x - 2) \ln (5) = (5 - 2 x) \ln (2) + \ln (7^{3})$

Step 3.3

Expand $\ln (7^{3})$ by moving $3$ outside the logarithm.

$4 \ln (3) + (3 x - 2) \ln (5) = (5 - 2 x) \ln (2) + 3 \ln (7)$

Step 4

Simplify the left side.

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Step 4.1

Apply the distributive property.

$4 \ln (3) + 3 x \ln (5) - 2 \ln (5) = (5 - 2 x) \ln (2) + 3 \ln (7)$

Step 5

Simplify the right side.

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Step 5.1

Apply the distributive property.

$4 \ln (3) + 3 x \ln (5) - 2 \ln (5) = 5 \ln (2) - 2 x \ln (2) + 3 \ln (7)$

Step 6

Reorder $4 \ln (3)$ and $3 x \ln (5)$ .

$3 x \ln (5) + 4 \ln (3) - 2 \ln (5) = 5 \ln (2) - 2 x \ln (2) + 3 \ln (7)$

Step 7

Reorder $5 \ln (2)$ and $- 2 x \ln (2)$ .

$3 x \ln (5) + 4 \ln (3) - 2 \ln (5) = - 2 x \ln (2) + 5 \ln (2) + 3 \ln (7)$

Step 8

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

$3 x \ln (5) + 4 \ln (3) - 2 \ln (5) + 2 x \ln (2) - 5 \ln (2) - 3 \ln (7) = 0$

Step 9

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

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Step 9.1

Subtract $4 \ln (3)$ from both sides of the equation.

$3 x \ln (5) - 2 \ln (5) + 2 x \ln (2) - 5 \ln (2) - 3 \ln (7) = - 4 \ln (3)$

Step 9.2

Add $2 \ln (5)$ to both sides of the equation.

$3 x \ln (5) + 2 x \ln (2) - 5 \ln (2) - 3 \ln (7) = - 4 \ln (3) + 2 \ln (5)$

Step 9.3

Add $5 \ln (2)$ to both sides of the equation.

$3 x \ln (5) + 2 x \ln (2) - 3 \ln (7) = - 4 \ln (3) + 2 \ln (5) + 5 \ln (2)$

Step 9.4

Add $3 \ln (7)$ to both sides of the equation.

$3 x \ln (5) + 2 x \ln (2) = - 4 \ln (3) + 2 \ln (5) + 5 \ln (2) + 3 \ln (7)$

Step 10

Factor $x$ out of $3 x \ln (5) + 2 x \ln (2)$ .

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Step 10.1

Factor $x$ out of $3 x \ln (5)$ .

$x (3 \ln (5)) + 2 x \ln (2) = - 4 \ln (3) + 2 \ln (5) + 5 \ln (2) + 3 \ln (7)$

Step 10.2

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

$x (3 \ln (5)) + x (2 \ln (2)) = - 4 \ln (3) + 2 \ln (5) + 5 \ln (2) + 3 \ln (7)$

Step 10.3

Factor $x$ out of $x (3 \ln (5)) + x (2 \ln (2))$ .

$x (3 \ln (5) + 2 \ln (2)) = - 4 \ln (3) + 2 \ln (5) + 5 \ln (2) + 3 \ln (7)$

Step 11

Divide each term in $x (3 \ln (5) + 2 \ln (2)) = - 4 \ln (3) + 2 \ln (5) + 5 \ln (2) + 3 \ln (7)$ by $3 \ln (5) + 2 \ln (2)$ and simplify.

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Step 11.1

Divide each term in $x (3 \ln (5) + 2 \ln (2)) = - 4 \ln (3) + 2 \ln (5) + 5 \ln (2) + 3 \ln (7)$ by $3 \ln (5) + 2 \ln (2)$ .

$\frac{x (3 \ln (5) + 2 \ln (2))}{3 \ln (5) + 2 \ln (2)} = \frac{- 4 \ln (3)}{3 \ln (5) + 2 \ln (2)} + \frac{2 \ln (5)}{3 \ln (5) + 2 \ln (2)} + \frac{5 \ln (2)}{3 \ln (5) + 2 \ln (2)} + \frac{3 \ln (7)}{3 \ln (5) + 2 \ln (2)}$

Step 11.2

Simplify the left side.

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Step 11.2.1

Cancel the common factor of $3 \ln (5) + 2 \ln (2)$ .

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Step 11.2.1.1

Cancel the common factor.

Step 11.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 4 \ln (3)}{3 \ln (5) + 2 \ln (2)} + \frac{2 \ln (5)}{3 \ln (5) + 2 \ln (2)} + \frac{5 \ln (2)}{3 \ln (5) + 2 \ln (2)} + \frac{3 \ln (7)}{3 \ln (5) + 2 \ln (2)}$

Step 11.3

Simplify the right side.

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Step 11.3.1

Simplify terms.

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Step 11.3.1.1

Move the negative in front of the fraction.

$x = - \frac{4 \ln (3)}{3 \ln (5) + 2 \ln (2)} + \frac{2 \ln (5)}{3 \ln (5) + 2 \ln (2)} + \frac{5 \ln (2)}{3 \ln (5) + 2 \ln (2)} + \frac{3 \ln (7)}{3 \ln (5) + 2 \ln (2)}$

Step 11.3.1.2

Combine the numerators over the common denominator.

$x = \frac{- 4 \ln (3) + 2 \ln (5)}{3 \ln (5) + 2 \ln (2)} + \frac{5 \ln (2)}{3 \ln (5) + 2 \ln (2)} + \frac{3 \ln (7)}{3 \ln (5) + 2 \ln (2)}$

Step 11.3.1.3

Factor $2$ out of $- 4 \ln (3) + 2 \ln (5)$ .

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Step 11.3.1.3.1

Factor $2$ out of $- 4 \ln (3)$ .

$x = \frac{2 (- 2 \ln (3)) + 2 \ln (5)}{3 \ln (5) + 2 \ln (2)} + \frac{5 \ln (2)}{3 \ln (5) + 2 \ln (2)} + \frac{3 \ln (7)}{3 \ln (5) + 2 \ln (2)}$

Step 11.3.1.3.2

Factor $2$ out of $2 \ln (5)$ .

$x = \frac{2 (- 2 \ln (3)) + 2 \ln (5)}{3 \ln (5) + 2 \ln (2)} + \frac{5 \ln (2)}{3 \ln (5) + 2 \ln (2)} + \frac{3 \ln (7)}{3 \ln (5) + 2 \ln (2)}$

Step 11.3.1.3.3

Factor $2$ out of $2 (- 2 \ln (3)) + 2 \ln (5)$ .

$x = \frac{2 (- 2 \ln (3) + \ln (5))}{3 \ln (5) + 2 \ln (2)} + \frac{5 \ln (2)}{3 \ln (5) + 2 \ln (2)} + \frac{3 \ln (7)}{3 \ln (5) + 2 \ln (2)}$

Step 11.3.1.4

Combine the numerators over the common denominator.

$x = \frac{2 (- 2 \ln (3) + \ln (5)) + 5 \ln (2)}{3 \ln (5) + 2 \ln (2)} + \frac{3 \ln (7)}{3 \ln (5) + 2 \ln (2)}$

Step 11.3.2

Simplify the numerator.

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Step 11.3.2.1

Apply the distributive property.

$x = \frac{2 (- 2 \ln (3)) + 2 \ln (5) + 5 \ln (2)}{3 \ln (5) + 2 \ln (2)} + \frac{3 \ln (7)}{3 \ln (5) + 2 \ln (2)}$

Step 11.3.2.2

Multiply $- 2$ by $2$ .

$x = \frac{- 4 \ln (3) + 2 \ln (5) + 5 \ln (2)}{3 \ln (5) + 2 \ln (2)} + \frac{3 \ln (7)}{3 \ln (5) + 2 \ln (2)}$

Step 11.3.3

Simplify terms.

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Step 11.3.3.1

Combine the numerators over the common denominator.

$x = \frac{- 4 \ln (3) + 2 \ln (5) + 5 \ln (2) + 3 \ln (7)}{3 \ln (5) + 2 \ln (2)}$

Step 11.3.3.2

Factor $- 1$ out of $- 4 \ln (3)$ .

$x = \frac{- (4 \ln (3)) + 2 \ln (5) + 5 \ln (2) + 3 \ln (7)}{3 \ln (5) + 2 \ln (2)}$

Step 11.3.3.3

Factor $- 1$ out of $2 \ln (5)$ .

$x = \frac{- (4 \ln (3)) - (- 2 \ln (5)) + 5 \ln (2) + 3 \ln (7)}{3 \ln (5) + 2 \ln (2)}$

Step 11.3.3.4

Factor $- 1$ out of $- (4 \ln (3)) - (- 2 \ln (5))$ .

$x = \frac{- (4 \ln (3) - 2 \ln (5)) + 5 \ln (2) + 3 \ln (7)}{3 \ln (5) + 2 \ln (2)}$

Step 11.3.3.5

Factor $- 1$ out of $5 \ln (2)$ .

$x = \frac{- (4 \ln (3) - 2 \ln (5)) - (- 5 \ln (2)) + 3 \ln (7)}{3 \ln (5) + 2 \ln (2)}$

Step 11.3.3.6

Factor $- 1$ out of $- (4 \ln (3) - 2 \ln (5)) - (- 5 \ln (2))$ .

$x = \frac{- (4 \ln (3) - 2 \ln (5) - 5 \ln (2)) + 3 \ln (7)}{3 \ln (5) + 2 \ln (2)}$

Step 11.3.3.7

Factor $- 1$ out of $3 \ln (7)$ .

$x = \frac{- (4 \ln (3) - 2 \ln (5) - 5 \ln (2)) - (- 3 \ln (7))}{3 \ln (5) + 2 \ln (2)}$

Step 11.3.3.8

Factor $- 1$ out of $- (4 \ln (3) - 2 \ln (5) - 5 \ln (2)) - (- 3 \ln (7))$ .

$x = \frac{- (4 \ln (3) - 2 \ln (5) - 5 \ln (2) - 3 \ln (7))}{3 \ln (5) + 2 \ln (2)}$

Step 11.3.3.9

Simplify the expression.

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Step 11.3.3.9.1

Rewrite $- (4 \ln (3) - 2 \ln (5) - 5 \ln (2) - 3 \ln (7))$ as $- 1 (4 \ln (3) - 2 \ln (5) - 5 \ln (2) - 3 \ln (7))$ .

$x = \frac{- 1 (4 \ln (3) - 2 \ln (5) - 5 \ln (2) - 3 \ln (7))}{3 \ln (5) + 2 \ln (2)}$

Step 11.3.3.9.2

Move the negative in front of the fraction.

$x = - \frac{4 \ln (3) - 2 \ln (5) - 5 \ln (2) - 3 \ln (7)}{3 \ln (5) + 2 \ln (2)}$

Step 12

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{4 \ln (3) - 2 \ln (5) - 5 \ln (2) - 3 \ln (7)}{3 \ln (5) + 2 \ln (2)}$

Decimal Form:

$x = 1.30786895 \dots$