Basic Math Examples

${(x - 3)}^{2 x - 6} = 1$

Step 1

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

$\ln ({(x - 3)}^{2 x - 6}) = \ln (1)$

Step 2

Expand $\ln ({(x - 3)}^{2 x - 6})$ by moving $2 x - 6$ outside the logarithm.

$(2 x - 6) \ln (x - 3) = \ln (1)$

Step 3

Simplify the left side.

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Step 3.1

Apply the distributive property.

$2 x \ln (x - 3) - 6 \ln (x - 3) = \ln (1)$

Step 4

Simplify the right side.

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Step 4.1

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

$2 x \ln (x - 3) - 6 \ln (x - 3) = 0$

Step 5

Factor $2 \ln (x - 3)$ out of $2 x \ln (x - 3) - 6 \ln (x - 3)$ .

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Step 5.1

Factor $2 \ln (x - 3)$ out of $2 x \ln (x - 3)$ .

$2 \ln (x - 3) x - 6 \ln (x - 3) = 0$

Step 5.2

Factor $2 \ln (x - 3)$ out of $- 6 \ln (x - 3)$ .

$2 \ln (x - 3) x + 2 \ln (x - 3) \cdot - 3 = 0$

Step 5.3

Factor $2 \ln (x - 3)$ out of $2 \ln (x - 3) x + 2 \ln (x - 3) \cdot - 3$ .

$2 \ln (x - 3) (x - 3) = 0$

Step 6

Divide each term in $2 \ln (x - 3) (x - 3) = 0$ by $2 (x - 3)$ and simplify.

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Step 6.1

Divide each term in $2 \ln (x - 3) (x - 3) = 0$ by $2 (x - 3)$ .

$\frac{2 \ln (x - 3) (x - 3)}{2 (x - 3)} = \frac{0}{2 (x - 3)}$

Step 6.2

Simplify the left side.

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Step 6.2.1

Cancel the common factor of $2$ .

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Step 6.2.1.1

Cancel the common factor.

$\frac{2 \ln (x - 3) (x - 3)}{2 (x - 3)} = \frac{0}{2 (x - 3)}$

Step 6.2.1.2

Rewrite the expression.

$\frac{\ln (x - 3) (x - 3)}{x - 3} = \frac{0}{2 (x - 3)}$

Step 6.2.2

Cancel the common factor of $x - 3$ .

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Step 6.2.2.1

Cancel the common factor.

$\frac{\ln (x - 3) (x - 3)}{x - 3} = \frac{0}{2 (x - 3)}$

Step 6.2.2.2

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

$\ln (x - 3) = \frac{0}{2 (x - 3)}$

Step 6.3

Simplify the right side.

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Step 6.3.1

Cancel the common factor of $0$ and $2$ .

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Step 6.3.1.1

Factor $2$ out of $0$ .

$\ln (x - 3) = \frac{2 (0)}{2 (x - 3)}$

Step 6.3.1.2

Cancel the common factors.

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Step 6.3.1.2.1

Cancel the common factor.

$\ln (x - 3) = \frac{2 \cdot 0}{2 (x - 3)}$

Step 6.3.1.2.2

Rewrite the expression.

$\ln (x - 3) = \frac{0}{x - 3}$

Step 6.3.2

Divide $0$ by $x - 3$ .

$\ln (x - 3) = 0$

Step 7

To solve for $x$ , rewrite the equation using properties of logarithms.

$e^{\ln (x - 3)} = e^{0}$

Step 8

Rewrite $\ln (x - 3) = 0$ in exponential form using the definition of a logarithm. If $x$ and $b$ are positive real numbers and $b \neq 1$ , then $\log_{b} (x) = y$ is equivalent to $b^{y} = x$ .

$e^{0} = x - 3$

Step 9

Solve for $x$ .

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Step 9.1

Rewrite the equation as $x - 3 = e^{0}$ .

$x - 3 = e^{0}$

Step 9.2

Anything raised to $0$ is $1$ .

$x - 3 = 1$

Step 9.3

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

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Step 9.3.1

Add $3$ to both sides of the equation.

$x = 1 + 3$

Step 9.3.2

Add $1$ and $3$ .

$x = 4$