Algebra Examples

$2^{x + 1} = 5^{1 - 2 x}$

Step 1

Take the log of both sides of the equation.

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

Step 2

Expand $\ln (2^{x + 1})$ by moving $x + 1$ outside the logarithm.

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

Step 3

Expand $\ln (5^{1 - 2 x})$ by moving $1 - 2 x$ outside the logarithm.

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

Step 4

Solve the equation for $x$ .

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Step 4.1

Simplify $(x + 1) \ln (2)$ .

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Step 4.1.1

Rewrite.

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

Step 4.1.2

Simplify by adding zeros.

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

Step 4.1.3

Apply the distributive property.

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

Step 4.1.4

Multiply $\ln (2)$ by $1$ .

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

Step 4.2

Simplify $(1 - 2 x) \ln (5)$ .

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Step 4.2.1

Apply the distributive property.

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

Step 4.2.2

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

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

Step 4.3

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

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

Step 4.4

Subtract $\ln (2)$ from both sides of the equation.

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

Step 4.5

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

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Step 4.5.1

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

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

Step 4.5.2

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

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

Step 4.5.3

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

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

Step 4.6

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

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Step 4.6.1

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

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

Step 4.6.2

Simplify the left side.

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Step 4.6.2.1

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

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Step 4.6.2.1.1

Cancel the common factor.

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

Step 4.6.2.1.2

Divide $x$ by $1$ .

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

Step 4.6.3

Simplify the right side.

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Step 4.6.3.1

Combine the numerators over the common denominator.

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

Step 5

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$x = 0.23422426 \dots$