Precalculus Examples

$f (x) = 3 \log (x + 5) - 2$

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 $x$ .

$0 = 3 \log (x + 5) - 2$

Step 1.2

Solve the equation.

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Step 1.2.1

Rewrite the equation as $3 \log (x + 5) - 2 = 0$ .

$3 \log (x + 5) - 2 = 0$

Step 1.2.2

Add $2$ to both sides of the equation.

$3 \log (x + 5) = 2$

Step 1.2.3

Divide each term in $3 \log (x + 5) = 2$ by $3$ and simplify.

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Step 1.2.3.1

Divide each term in $3 \log (x + 5) = 2$ by $3$ .

$\frac{3 \log (x + 5)}{3} = \frac{2}{3}$

Step 1.2.3.2

Simplify the left side.

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Step 1.2.3.2.1

Cancel the common factor of $3$ .

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Step 1.2.3.2.1.1

Cancel the common factor.

$\frac{3 \log (x + 5)}{3} = \frac{2}{3}$

Step 1.2.3.2.1.2

Divide $\log (x + 5)$ by $1$ .

$\log (x + 5) = \frac{2}{3}$

Step 1.2.4

Rewrite $\log (x + 5) = \frac{2}{3}$ 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 $b^{y} = x$ .

$10^{\frac{2}{3}} = x + 5$

Step 1.2.5

Solve for $x$ .

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Step 1.2.5.1

Rewrite the equation as $x + 5 = 10^{\frac{2}{3}}$ .

$x + 5 = 10^{\frac{2}{3}}$

Step 1.2.5.2

Subtract $5$ from both sides of the equation.

$x = 10^{\frac{2}{3}} - 5$

Step 1.3

x-intercept(s) in point form.

x-intercept(s): $(10^{\frac{2}{3}} - 5, 0)$

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 $y$ .

$y = 3 \log ((0) + 5) - 2$

Step 2.2

Solve the equation.

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Step 2.2.1

Remove parentheses.

$y = 3 \log (0 + 5) - 2$

Step 2.2.2

Remove parentheses.

$y = 3 \log ((0) + 5) - 2$

Step 2.2.3

Simplify each term.

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Step 2.2.3.1

Add $0$ and $5$ .

$y = 3 \log (5) - 2$

Step 2.2.3.2

Simplify $3 \log (5)$ by moving $3$ inside the logarithm.

$y = \log (5^{3}) - 2$

Step 2.2.3.3

Raise $5$ to the power of $3$ .

$y = \log (125) - 2$

Step 2.3

y-intercept(s) in point form.

y-intercept(s): $(0, \log (125) - 2)$

Step 3

List the intersections.

x-intercept(s): $(10^{\frac{2}{3}} - 5, 0)$

y-intercept(s): $(0, \log (125) - 2)$

Step 4