Algebra Examples

$4^{x + 4} = 5^{2 x + 5}$

Step 1

Take the log of both sides of the equation.

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

Step 2

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

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

Step 3

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

$(x + 4) \ln (4) = (2 x + 5) \ln (5)$

Step 4

Solve the equation for $x$ .

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Step 4.1

Simplify $(x + 4) \ln (4)$ .

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Step 4.1.1

Rewrite.

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

Step 4.1.2

Simplify by adding zeros.

$(x + 4) \ln (4) = (2 x + 5) \ln (5)$

Step 4.1.3

Apply the distributive property.

$x \ln (4) + 4 \ln (4) = (2 x + 5) \ln (5)$

Step 4.2

Apply the distributive property.

$x \ln (4) + 4 \ln (4) = 2 x \ln (5) + 5 \ln (5)$

Step 4.3

Subtract $2 x \ln (5)$ from both sides of the equation.

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

Step 4.4

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

$x \ln (4) - 2 x \ln (5) = 5 \ln (5) - 4 \ln (4)$

Step 4.5

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

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Step 4.5.1

Factor $x$ out of $x \ln (4)$ .

$x (\ln (4)) - 2 x \ln (5) = 5 \ln (5) - 4 \ln (4)$

Step 4.5.2

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

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

Step 4.5.3

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

$x (\ln (4) - 2 \ln (5)) = 5 \ln (5) - 4 \ln (4)$

Step 4.6

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

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Step 4.6.1

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

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

Step 4.6.2

Simplify the left side.

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Step 4.6.2.1

Cancel the common factor of $\ln (4) - 2 \ln (5)$ .

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Step 4.6.2.1.1

Cancel the common factor.

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

Step 4.6.2.1.2

Divide $x$ by $1$ .

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

Step 4.6.3

Simplify the right side.

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Step 4.6.3.1

Combine the numerators over the common denominator.

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

Step 5

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$x = - 1.36529380 \dots$