Precalculus Examples

${(\frac{2}{3})}^{x} = 5^{1 - x}$

Step 1

Take the log of both sides of the equation.

$\ln ({(\frac{2}{3})}^{x}) = \ln (5^{1 - x})$

Step 2

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

$x \ln (\frac{2}{3}) = \ln (5^{1 - x})$

Step 3

Rewrite $\ln (\frac{2}{3})$ as $\ln (2) - \ln (3)$ .

$x (\ln (2) - \ln (3)) = \ln (5^{1 - x})$

Step 4

Expand $\ln (5^{1 - x})$ by moving $1 - x$ outside the logarithm.

$x (\ln (2) - \ln (3)) = (1 - x) \ln (5)$

Step 5

Solve the equation for $x$ .

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Step 5.1

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

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Step 5.1.1

Rewrite.

$0 + 0 + x (\ln (2) - \ln (3)) = (1 - x) \ln (5)$

Step 5.1.2

Simplify by adding zeros.

$x (\ln (2) - \ln (3)) = (1 - x) \ln (5)$

Step 5.1.3

Apply the distributive property.

$x \ln (2) + x (- \ln (3)) = (1 - x) \ln (5)$

Step 5.1.4

Rewrite using the commutative property of multiplication.

$x \ln (2) - x \ln (3) = (1 - x) \ln (5)$

Step 5.2

Simplify $(1 - x) \ln (5)$ .

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Step 5.2.1

Apply the distributive property.

$x \ln (2) - x \ln (3) = 1 \ln (5) - x \ln (5)$

Step 5.2.2

Multiply $\ln (5)$ by $1$ .

$x \ln (2) - x \ln (3) = \ln (5) - x \ln (5)$

Step 5.3

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

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

Step 5.4

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

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Step 5.4.1

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

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

Step 5.4.2

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

$x (\ln (2)) + x (- 1 \ln (3)) + x \ln (5) = \ln (5)$

Step 5.4.3

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

$x (\ln (2)) + x (- 1 \ln (3)) + x (\ln (5)) = \ln (5)$

Step 5.4.4

Factor $x$ out of $x (\ln (2)) + x (- 1 \ln (3))$ .

$x (\ln (2) - 1 \ln (3)) + x (\ln (5)) = \ln (5)$

Step 5.4.5

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

$x (\ln (2) - 1 \ln (3) + \ln (5)) = \ln (5)$

Step 5.5

Rewrite $- 1 \ln (3)$ as $- \ln (3)$ .

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

Step 5.6

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

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Step 5.6.1

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

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

Step 5.6.2

Simplify the left side.

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Step 5.6.2.1

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

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Step 5.6.2.1.1

Cancel the common factor.

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

Step 5.6.2.1.2

Divide $x$ by $1$ .

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

Step 6

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$x = 1.33677264 \dots$