Algebra Examples

$4^{3 + x} = 25$

Step 1

Take the natural logarithm of both sides of the equation to remove the variable from the exponent.

$\ln (4^{3 + x}) = \ln (25)$

Step 2

Expand $\ln (4^{3 + x})$ by moving $3 + x$ outside the logarithm.

$(3 + x) \ln (4) = \ln (25)$

Step 3

Simplify the left side.

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Step 3.1

Apply the distributive property.

$3 \ln (4) + x \ln (4) = \ln (25)$

Step 4

Reorder $3 \ln (4)$ and $x \ln (4)$ .

$x \ln (4) + 3 \ln (4) = \ln (25)$

Step 5

Move all the terms containing a logarithm to the left side of the equation.

$x \ln (4) + 3 \ln (4) - \ln (25) = 0$

Step 6

Move all terms not containing $x$ to the right side of the equation.

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Step 6.1

Subtract $3 \ln (4)$ from both sides of the equation.

$x \ln (4) - \ln (25) = - 3 \ln (4)$

Step 6.2

Add $\ln (25)$ to both sides of the equation.

$x \ln (4) = - 3 \ln (4) + \ln (25)$

Step 7

Divide each term in $x \ln (4) = - 3 \ln (4) + \ln (25)$ by $\ln (4)$ and simplify.

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Step 7.1

Divide each term in $x \ln (4) = - 3 \ln (4) + \ln (25)$ by $\ln (4)$ .

$\frac{x \ln (4)}{\ln (4)} = \frac{- 3 \ln (4)}{\ln (4)} + \frac{\ln (25)}{\ln (4)}$

Step 7.2

Simplify the left side.

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Step 7.2.1

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

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Step 7.2.1.1

Cancel the common factor.

$\frac{x \ln (4)}{\ln (4)} = \frac{- 3 \ln (4)}{\ln (4)} + \frac{\ln (25)}{\ln (4)}$

Step 7.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 3 \ln (4)}{\ln (4)} + \frac{\ln (25)}{\ln (4)}$

Step 7.3

Simplify the right side.

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Step 7.3.1

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

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Step 7.3.1.1

Cancel the common factor.

$x = \frac{- 3 \ln (4)}{\ln (4)} + \frac{\ln (25)}{\ln (4)}$

Step 7.3.1.2

Divide $- 3$ by $1$ .

$x = - 3 + \frac{\ln (25)}{\ln (4)}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$x = - 3 + \frac{\ln (25)}{\ln (4)}$

Decimal Form:

$x = - 0.67807190 \dots$