Algebra Examples

$f (x) = \ln (- 4 x) + 3$

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 $x$ .

$0 = \ln (- 4 x) + 3$

Step 1.2

Solve the equation.

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Step 1.2.1

Rewrite the equation as $\ln (- 4 x) + 3 = 0$ .

$\ln (- 4 x) + 3 = 0$

Step 1.2.2

Subtract $3$ from both sides of the equation.

$\ln (- 4 x) = - 3$

Step 1.2.3

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

$e^{\ln (- 4 x)} = e^{- 3}$

Step 1.2.4

Rewrite $\ln (- 4 x) = - 3$ 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 $b^{y} = x$ .

$e^{- 3} = - 4 x$

Step 1.2.5

Solve for $x$ .

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Step 1.2.5.1

Rewrite the equation as $- 4 x = e^{- 3}$ .

$- 4 x = e^{- 3}$

Step 1.2.5.2

Divide each term in $- 4 x = e^{- 3}$ by $- 4$ and simplify.

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Step 1.2.5.2.1

Divide each term in $- 4 x = e^{- 3}$ by $- 4$ .

$\frac{- 4 x}{- 4} = \frac{e^{- 3}}{- 4}$

Step 1.2.5.2.2

Simplify the left side.

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Step 1.2.5.2.2.1

Cancel the common factor of $- 4$ .

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Step 1.2.5.2.2.1.1

Cancel the common factor.

$\frac{- 4 x}{- 4} = \frac{e^{- 3}}{- 4}$

Step 1.2.5.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{e^{- 3}}{- 4}$

Step 1.2.5.2.3

Simplify the right side.

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Step 1.2.5.2.3.1

Move $e^{- 3}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$x = \frac{1}{- 4 e^{3}}$

Step 1.2.5.2.3.2

Move the negative in front of the fraction.

$x = - \frac{1}{4 e^{3}}$

Step 1.3

x-intercept(s) in point form.

x-intercept(s): $(- \frac{1}{4 e^{3}}, 0)$

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 $y$ .

$y = \ln (- 4 \cdot 0) + 3$

Step 2.2

Solve the equation.

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Step 2.2.1

Remove parentheses.

$y = \ln (- 4 \cdot 0) + 3$

Step 2.2.2

Simplify the right side.

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Step 2.2.2.1

Multiply $- 4$ by $0$ .

$y = \ln (0) + 3$

Step 2.2.2.2

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): $(- \frac{1}{4 e^{3}}, 0)$

y-intercept(s): $None$

Step 4