Algebra Examples

$\frac{1}{3} (13^{6 x + 5}) = 14$

Step 1

Multiply both sides of the equation by $3$ .

$3 (\frac{1}{3} \cdot 13^{6 x + 5}) = 3 \cdot 14$

Step 2

Simplify both sides of the equation.

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Step 2.1

Simplify the left side.

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Step 2.1.1

Simplify $3 (\frac{1}{3} \cdot 13^{6 x + 5})$ .

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Step 2.1.1.1

Combine $\frac{1}{3}$ and $13^{6 x + 5}$ .

$3 \frac{13^{6 x + 5}}{3} = 3 \cdot 14$

Step 2.1.1.2

Cancel the common factor of $3$ .

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Step 2.1.1.2.1

Cancel the common factor.

$3 \frac{13^{6 x + 5}}{3} = 3 \cdot 14$

Step 2.1.1.2.2

Rewrite the expression.

$13^{6 x + 5} = 3 \cdot 14$

Step 2.2

Simplify the right side.

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Step 2.2.1

Multiply $3$ by $14$ .

$13^{6 x + 5} = 42$

Step 3

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

$\ln (13^{6 x + 5}) = \ln (42)$

Step 4

Expand $\ln (13^{6 x + 5})$ by moving $6 x + 5$ outside the logarithm.

$(6 x + 5) \ln (13) = \ln (42)$

Step 5

Simplify the left side.

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Step 5.1

Apply the distributive property.

$6 x \ln (13) + 5 \ln (13) = \ln (42)$

Step 6

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

$6 x \ln (13) + 5 \ln (13) - \ln (42) = 0$

Step 7

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

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Step 7.1

Subtract $5 \ln (13)$ from both sides of the equation.

$6 x \ln (13) - \ln (42) = - 5 \ln (13)$

Step 7.2

Add $\ln (42)$ to both sides of the equation.

$6 x \ln (13) = - 5 \ln (13) + \ln (42)$

Step 8

Divide each term in $6 x \ln (13) = - 5 \ln (13) + \ln (42)$ by $6 \ln (13)$ and simplify.

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Step 8.1

Divide each term in $6 x \ln (13) = - 5 \ln (13) + \ln (42)$ by $6 \ln (13)$ .

$\frac{6 x \ln (13)}{6 \ln (13)} = \frac{- 5 \ln (13)}{6 \ln (13)} + \frac{\ln (42)}{6 \ln (13)}$

Step 8.2

Simplify the left side.

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Step 8.2.1

Cancel the common factor of $6$ .

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Step 8.2.1.1

Cancel the common factor.

$\frac{6 x \ln (13)}{6 \ln (13)} = \frac{- 5 \ln (13)}{6 \ln (13)} + \frac{\ln (42)}{6 \ln (13)}$

Step 8.2.1.2

Rewrite the expression.

$\frac{x \ln (13)}{\ln (13)} = \frac{- 5 \ln (13)}{6 \ln (13)} + \frac{\ln (42)}{6 \ln (13)}$

Step 8.2.2

Cancel the common factor of $\ln (13)$ .

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Step 8.2.2.1

Cancel the common factor.

$\frac{x \ln (13)}{\ln (13)} = \frac{- 5 \ln (13)}{6 \ln (13)} + \frac{\ln (42)}{6 \ln (13)}$

Step 8.2.2.2

Divide $x$ by $1$ .

$x = \frac{- 5 \ln (13)}{6 \ln (13)} + \frac{\ln (42)}{6 \ln (13)}$

Step 8.3

Simplify the right side.

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Step 8.3.1

Simplify each term.

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Step 8.3.1.1

Cancel the common factor of $\ln (13)$ .

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Step 8.3.1.1.1

Cancel the common factor.

$x = \frac{- 5 \ln (13)}{6 \ln (13)} + \frac{\ln (42)}{6 \ln (13)}$

Step 8.3.1.1.2

Rewrite the expression.

$x = \frac{- 5}{6} + \frac{\ln (42)}{6 \ln (13)}$

Step 8.3.1.2

Move the negative in front of the fraction.

$x = - \frac{5}{6} + \frac{\ln (42)}{6 \ln (13)}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{5}{6} + \frac{\ln (42)}{6 \ln (13)}$

Decimal Form:

$x = - 0.59046501 \dots$