Algebra Examples

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

Step 1

Rewrite the equation as $5^{2 x + 1} = 1$ .

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

Step 2

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

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

Step 3

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

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

Step 4

Simplify the left side.

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Step 4.1

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

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Step 4.1.1

Apply the distributive property.

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

Step 4.1.2

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

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

Step 5

Simplify the right side.

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Step 5.1

The natural logarithm of $1$ is $0$ .

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

Step 6

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

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

Step 7

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

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Step 7.1

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

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

Step 7.2

Simplify the left side.

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Step 7.2.1

Cancel the common factor of $2$ .

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Step 7.2.1.1

Cancel the common factor.

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

Step 7.2.1.2

Rewrite the expression.

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

Step 7.2.2

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

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Step 7.2.2.1

Cancel the common factor.

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

Step 7.2.2.2

Divide $x$ by $1$ .

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

Step 7.3

Simplify the right side.

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Step 7.3.1

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

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Step 7.3.1.1

Cancel the common factor.

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

Step 7.3.1.2

Rewrite the expression.

$x = \frac{- 1}{2}$

Step 7.3.2

Move the negative in front of the fraction.

$x = - \frac{1}{2}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{1}{2}$

Decimal Form:

$x = - 0.5$