Algebra Examples

${(\frac{3}{5})}^{x} - 7^{1 - x} = 0$

Step 1

Move $- 7^{1 - x}$ to the right side of the equation by adding it to both sides.

${(\frac{3}{5})}^{x} = 7^{1 - x}$

Step 2

Take the log of both sides of the equation.

$\ln ({(\frac{3}{5})}^{x}) = \ln (7^{1 - x})$

Step 3

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

$x \ln (\frac{3}{5}) = \ln (7^{1 - x})$

Step 4

Rewrite $\ln (\frac{3}{5})$ as $\ln (3) - \ln (5)$ .

$x (\ln (3) - \ln (5)) = \ln (7^{1 - x})$

Step 5

Expand $\ln (7^{1 - x})$ by moving $1 - x$ outside the logarithm.

$x (\ln (3) - \ln (5)) = (1 - x) \ln (7)$

Step 6

Solve the equation for $x$ .

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Step 6.1

Simplify $x (\ln (3) - \ln (5))$ .

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Step 6.1.1

Rewrite.

$0 + 0 + x (\ln (3) - \ln (5)) = (1 - x) \ln (7)$

Step 6.1.2

Simplify by adding zeros.

$x (\ln (3) - \ln (5)) = (1 - x) \ln (7)$

Step 6.1.3

Apply the distributive property.

$x \ln (3) + x (- \ln (5)) = (1 - x) \ln (7)$

Step 6.1.4

Rewrite using the commutative property of multiplication.

$x \ln (3) - x \ln (5) = (1 - x) \ln (7)$

Step 6.2

Simplify $(1 - x) \ln (7)$ .

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Step 6.2.1

Apply the distributive property.

$x \ln (3) - x \ln (5) = 1 \ln (7) - x \ln (7)$

Step 6.2.2

Multiply $\ln (7)$ by $1$ .

$x \ln (3) - x \ln (5) = \ln (7) - x \ln (7)$

Step 6.3

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

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

Step 6.4

Factor $x$ out of $x \ln (3) - x \ln (5) + x \ln (7)$ .

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Step 6.4.1

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

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

Step 6.4.2

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

$x (\ln (3)) + x (- 1 \ln (5)) + x \ln (7) = \ln (7)$

Step 6.4.3

Factor $x$ out of $x \ln (7)$ .

$x (\ln (3)) + x (- 1 \ln (5)) + x (\ln (7)) = \ln (7)$

Step 6.4.4

Factor $x$ out of $x (\ln (3)) + x (- 1 \ln (5))$ .

$x (\ln (3) - 1 \ln (5)) + x (\ln (7)) = \ln (7)$

Step 6.4.5

Factor $x$ out of $x (\ln (3) - 1 \ln (5)) + x (\ln (7))$ .

$x (\ln (3) - 1 \ln (5) + \ln (7)) = \ln (7)$

Step 6.5

Rewrite $- 1 \ln (5)$ as $- \ln (5)$ .

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

Step 6.6

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

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Step 6.6.1

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

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

Step 6.6.2

Simplify the left side.

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Step 6.6.2.1

Cancel the common factor of $\ln (3) - \ln (5) + \ln (7)$ .

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Step 6.6.2.1.1

Cancel the common factor.

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

Step 6.6.2.1.2

Divide $x$ by $1$ .

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

Step 7

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$x = 1.35595507 \dots$