Algebra Examples

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

Step 1

Take the log of both sides of the equation.

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

Step 2

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

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

Step 3

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

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

Step 4

Solve the equation for $x$ .

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Step 4.1

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

$x \ln (4) - x^{2} \ln (5) = 0$

Step 4.2

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

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Step 4.2.1

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

$x (\ln (4)) - x^{2} \ln (5) = 0$

Step 4.2.2

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

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

Step 4.2.3

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

$x (\ln (4) - x \ln (5)) = 0$

Step 4.3

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x = 0$

$\ln (4) - x \ln (5) = 0$

Step 4.4

Set $x$ equal to $0$ .

$x = 0$

Step 4.5

Set $\ln (4) - x \ln (5)$ equal to $0$ and solve for $x$ .

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Step 4.5.1

Set $\ln (4) - x \ln (5)$ equal to $0$ .

$\ln (4) - x \ln (5) = 0$

Step 4.5.2

Solve $\ln (4) - x \ln (5) = 0$ for $x$ .

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Step 4.5.2.1

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

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

Step 4.5.2.2

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

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Step 4.5.2.2.1

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

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

Step 4.5.2.2.2

Simplify the left side.

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Step 4.5.2.2.2.1

Dividing two negative values results in a positive value.

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

Step 4.5.2.2.2.2

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

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Step 4.5.2.2.2.2.1

Cancel the common factor.

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

Step 4.5.2.2.2.2.2

Divide $x$ by $1$ .

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

Step 4.5.2.2.3

Simplify the right side.

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Step 4.5.2.2.3.1

Dividing two negative values results in a positive value.

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

Step 4.6

The final solution is all the values that make $x (\ln (4) - x \ln (5)) = 0$ true.

$x = 0, \frac{\ln (4)}{\ln (5)}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = 0, \frac{\ln (4)}{\ln (5)}$

Decimal Form:

$x = 0, 0.86135311 \dots$