Algebra Examples

3^{x - 1} = 5

$3^{x - 1} = 5$

Step 1

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

\ln (3^{x - 1}) = \ln (5)

$\ln (3^{x - 1}) = \ln (5)$

Step 2

Expand

\ln (3^{x - 1})

$\ln (3^{x - 1})$ by moving

x - 1

$x - 1$ outside the logarithm.

(x - 1) \ln (3) = \ln (5)

$(x - 1) \ln (3) = \ln (5)$

Step 3

Simplify the left side.

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Step 3.1

Simplify

(x - 1) \ln (3)

$(x - 1) \ln (3)$ .

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Step 3.1.1

Apply the distributive property.

x \ln (3) - 1 \ln (3) = \ln (5)

$x \ln (3) - 1 \ln (3) = \ln (5)$

Step 3.1.2

Rewrite

- 1 \ln (3)

$- 1 \ln (3)$ as

- \ln (3)

$- \ln (3)$ .

x \ln (3) - \ln (3) = \ln (5)

$x \ln (3) - \ln (3) = \ln (5)$

x \ln (3) - \ln (3) = \ln (5)

$x \ln (3) - \ln (3) = \ln (5)$

x \ln (3) - \ln (3) = \ln (5)

$x \ln (3) - \ln (3) = \ln (5)$

Step 4

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

x \ln (3) - \ln (3) - \ln (5) = 0

$x \ln (3) - \ln (3) - \ln (5) = 0$

Step 5

Move all terms not containing

x

$x$ to the right side of the equation.

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Step 5.1

Add

\ln (3)

$\ln (3)$ to both sides of the equation.

x \ln (3) - \ln (5) = \ln (3)

$x \ln (3) - \ln (5) = \ln (3)$

Step 5.2

Add

\ln (5)

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

x \ln (3) = \ln (3) + \ln (5)

$x \ln (3) = \ln (3) + \ln (5)$

x \ln (3) = \ln (3) + \ln (5)

$x \ln (3) = \ln (3) + \ln (5)$

Step 6

Divide each term in

x \ln (3) = \ln (3) + \ln (5)

$x \ln (3) = \ln (3) + \ln (5)$ by

\ln (3)

$\ln (3)$ and simplify.

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Step 6.1

Divide each term in

x \ln (3) = \ln (3) + \ln (5)

$x \ln (3) = \ln (3) + \ln (5)$ by

\ln (3)

$\ln (3)$ .

\frac{x \ln (3)}{\ln (3)} = \frac{\ln (3)}{\ln (3)} + \frac{\ln (5)}{\ln (3)}

$\frac{x \ln (3)}{\ln (3)} = \frac{\ln (3)}{\ln (3)} + \frac{\ln (5)}{\ln (3)}$

Step 6.2

Simplify the left side.

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Step 6.2.1

Cancel the common factor of

\ln (3)

$\ln (3)$ .

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Step 6.2.1.1

Cancel the common factor.

\frac{x \ln (3)}{\ln (3)} = \frac{\ln (3)}{\ln (3)} + \frac{\ln (5)}{\ln (3)}

$\frac{x \ln (3)}{\ln (3)} = \frac{\ln (3)}{\ln (3)} + \frac{\ln (5)}{\ln (3)}$

Step 6.2.1.2

Divide

x

$x$ by

1

$1$ .

x = \frac{\ln (3)}{\ln (3)} + \frac{\ln (5)}{\ln (3)}

$x = \frac{\ln (3)}{\ln (3)} + \frac{\ln (5)}{\ln (3)}$

x = \frac{\ln (3)}{\ln (3)} + \frac{\ln (5)}{\ln (3)}

$x = \frac{\ln (3)}{\ln (3)} + \frac{\ln (5)}{\ln (3)}$

x = \frac{\ln (3)}{\ln (3)} + \frac{\ln (5)}{\ln (3)}

$x = \frac{\ln (3)}{\ln (3)} + \frac{\ln (5)}{\ln (3)}$

Step 6.3

Simplify the right side.

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Step 6.3.1

Cancel the common factor of

\ln (3)

$\ln (3)$ .

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Step 6.3.1.1

Cancel the common factor.

x = \frac{\ln (3)}{\ln (3)} + \frac{\ln (5)}{\ln (3)}

$x = \frac{\ln (3)}{\ln (3)} + \frac{\ln (5)}{\ln (3)}$

Step 6.3.1.2

Rewrite the expression.

x = 1 + \frac{\ln (5)}{\ln (3)}

$x = 1 + \frac{\ln (5)}{\ln (3)}$

x = 1 + \frac{\ln (5)}{\ln (3)}

$x = 1 + \frac{\ln (5)}{\ln (3)}$

x = 1 + \frac{\ln (5)}{\ln (3)}

$x = 1 + \frac{\ln (5)}{\ln (3)}$

x = 1 + \frac{\ln (5)}{\ln (3)}

$x = 1 + \frac{\ln (5)}{\ln (3)}$

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