Precalculus Examples

$y = \log_{5} (1 - x)$

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 $x$ .

$0 = \log_{5} (1 - x)$

Step 1.2

Solve the equation.

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Step 1.2.1

Rewrite the equation as $\log_{5} (1 - x) = 0$ .

$\log_{5} (1 - x) = 0$

Step 1.2.2

Rewrite $\log_{5} (1 - x) = 0$ 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 $b^{y} = x$ .

$5^{0} = 1 - x$

Step 1.2.3

Solve for $x$ .

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Step 1.2.3.1

Rewrite the equation as $1 - x = 5^{0}$ .

$1 - x = 5^{0}$

Step 1.2.3.2

Anything raised to $0$ is $1$ .

$1 - x = 1$

Step 1.2.3.3

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

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Step 1.2.3.3.1

Subtract $1$ from both sides of the equation.

$- x = 1 - 1$

Step 1.2.3.3.2

Subtract $1$ from $1$ .

$- x = 0$

Step 1.2.3.4

Divide each term in $- x = 0$ by $- 1$ and simplify.

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Step 1.2.3.4.1

Divide each term in $- x = 0$ by $- 1$ .

$\frac{- x}{- 1} = \frac{0}{- 1}$

Step 1.2.3.4.2

Simplify the left side.

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Step 1.2.3.4.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} = \frac{0}{- 1}$

Step 1.2.3.4.2.2

Divide $x$ by $1$ .

$x = \frac{0}{- 1}$

Step 1.2.3.4.3

Simplify the right side.

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Step 1.2.3.4.3.1

Divide $0$ by $- 1$ .

$x = 0$

Step 1.3

x-intercept(s) in point form.

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

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 $y$ .

$y = \log_{5} (1 - (0))$

Step 2.2

Solve the equation.

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Step 2.2.1

Remove parentheses.

$y = \log_{5} (1 - (0))$

Step 2.2.2

Simplify $\log_{5} (1 - (0))$ .

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Step 2.2.2.1

Subtract $0$ from $1$ .

$y = \log_{5} (1)$

Step 2.2.2.2

Logarithm base $5$ of $1$ is $0$ .

$y = 0$

Step 2.3

y-intercept(s) in point form.

y-intercept(s): $(0, 0)$

Step 3

List the intersections.

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

y-intercept(s): $(0, 0)$

Step 4