Precalculus Examples

$y = - \log_{3} (x - 3) + 1$

Step 1

Find the x-intercepts.

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Step 1.1

To find the x-intercept(s), substitute in $0$ for $y$ and solve for $x$ .

$0 = - \log_{3} (x - 3) + 1$

Step 1.2

Solve the equation.

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Step 1.2.1

Rewrite the equation as $- \log_{3} (x - 3) + 1 = 0$ .

$- \log_{3} (x - 3) + 1 = 0$

Step 1.2.2

Subtract $1$ from both sides of the equation.

$- \log_{3} (x - 3) = - 1$

Step 1.2.3

Divide each term in $- \log_{3} (x - 3) = - 1$ by $- 1$ and simplify.

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Step 1.2.3.1

Divide each term in $- \log_{3} (x - 3) = - 1$ by $- 1$ .

$\frac{- \log_{3} (x - 3)}{- 1} = \frac{- 1}{- 1}$

Step 1.2.3.2

Simplify the left side.

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Step 1.2.3.2.1

Dividing two negative values results in a positive value.

$\frac{\log_{3} (x - 3)}{1} = \frac{- 1}{- 1}$

Step 1.2.3.2.2

Divide $\log_{3} (x - 3)$ by $1$ .

$\log_{3} (x - 3) = \frac{- 1}{- 1}$

Step 1.2.3.3

Simplify the right side.

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Step 1.2.3.3.1

Divide $- 1$ by $- 1$ .

$\log_{3} (x - 3) = 1$

Step 1.2.4

Rewrite $\log_{3} (x - 3) = 1$ 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 $b^{y} = x$ .

$3^{1} = x - 3$

Step 1.2.5

Solve for $x$ .

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Step 1.2.5.1

Rewrite the equation as $x - 3 = 3^{1}$ .

$x - 3 = 3$

Step 1.2.5.2

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

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Step 1.2.5.2.1

Add $3$ to both sides of the equation.

$x = 3 + 3$

Step 1.2.5.2.2

Add $3$ and $3$ .

$x = 6$

Step 1.3

x-intercept(s) in point form.

x-intercept(s): $(6, 0)$

Step 2

Find the y-intercepts.

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Step 2.1

To find the y-intercept(s), substitute in $0$ for $x$ and solve for $y$ .

$y = - \log_{3} ((0) - 3) + 1$

Step 2.2

Solve the equation.

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Step 2.2.1

The logarithm of a negative number is undefined.

$y = Undefined$

Step 2.2.2

Remove parentheses.

$y = - \log_{3} ((0) - 3) + 1$

Step 2.2.3

The equation cannot be solved because it is undefined.

$Undefined$

Step 2.3

To find the y-intercept(s), substitute in $0$ for $x$ and solve for $y$ .

y-intercept(s): $None$

Step 3

List the intersections.

x-intercept(s): $(6, 0)$

y-intercept(s): $None$

Step 4