Precalculus Examples

3 log (x + 5) + 2

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

Step 1

Write

3 log (x + 5) + 2

$3 \log (x + 5) + 2$ as an equation.

y = 3 log (x + 5) + 2

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

Step 2

Find the x-intercepts.

Tap for more steps...

Step 2.1

To find the x-intercept(s), substitute in

0

$0$ for

y

$y$ and solve for

x

$x$ .

0 = 3 log (x + 5) + 2

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

Step 2.2

Solve the equation.

Tap for more steps...

Step 2.2.1

Rewrite the equation as

3 log (x + 5) + 2 = 0

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

3 log (x + 5) + 2 = 0

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

Step 2.2.2

Subtract

2

$2$ from both sides of the equation.

3 log (x + 5) = - 2

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

Step 2.2.3

Divide each term in

3 log (x + 5) = - 2

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

3

$3$ and simplify.

Tap for more steps...

Step 2.2.3.1

Divide each term in

3 log (x + 5) = - 2

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

3

$3$ .

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

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

Step 2.2.3.2

Simplify the left side.

Tap for more steps...

Step 2.2.3.2.1

Cancel the common factor of

3

$3$ .

Tap for more steps...

Step 2.2.3.2.1.1

Cancel the common factor.

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

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

Step 2.2.3.2.1.2

Divide

log (x + 5)

$\log (x + 5)$ by

1

$1$ .

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

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

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

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

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

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

Step 2.2.3.3

Simplify the right side.

Tap for more steps...

Step 2.2.3.3.1

Move the negative in front of the fraction.

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

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

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

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

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

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

Step 2.2.4

Rewrite

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

$\log (x + 5) = - \frac{2}{3}$ in exponential form using the definition of a logarithm. If

x

$x$ and

b

$b$ are positive real numbers and

b \neq 1

$b \neq 1$ , then

{log}_{b} (x) = y

$\log_{b} (x) = y$ is equivalent to

b^{y} = x

$b^{y} = x$ .

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

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

Step 2.2.5

Solve for

x

$x$ .

Tap for more steps...

Step 2.2.5.1

Rewrite the equation as

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

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

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

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

Step 2.2.5.2

Rewrite the expression using the negative exponent rule

b^{- n} = \frac{1}{b^{n}}

$b^{- n} = \frac{1}{b^{n}}$ .

x + 5 = \frac{1}{10^{\frac{2}{3}}}

$x + 5 = \frac{1}{10^{\frac{2}{3}}}$

Step 2.2.5.3

Subtract

5

$5$ from both sides of the equation.

x = \frac{1}{10^{\frac{2}{3}}} - 5

$x = \frac{1}{10^{\frac{2}{3}}} - 5$

x = \frac{1}{10^{\frac{2}{3}}} - 5

$x = \frac{1}{10^{\frac{2}{3}}} - 5$

x = \frac{1}{10^{\frac{2}{3}}} - 5

$x = \frac{1}{10^{\frac{2}{3}}} - 5$

Step 2.3

x-intercept(s) in point form.

x-intercept(s):

(\frac{1}{10^{\frac{2}{3}}} - 5, 0)

$(\frac{1}{10^{\frac{2}{3}}} - 5, 0)$

x-intercept(s):

(\frac{1}{10^{\frac{2}{3}}} - 5, 0)

$(\frac{1}{10^{\frac{2}{3}}} - 5, 0)$

Step 3

Find the y-intercepts.

Tap for more steps...

Step 3.1

To find the y-intercept(s), substitute in

0

$0$ for

x

$x$ and solve for

y

$y$ .

y = 3 log ((0) + 5) + 2

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

Step 3.2

Solve the equation.

Tap for more steps...

Step 3.2.1

Remove parentheses.

y = 3 log (0 + 5) + 2

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

Step 3.2.2

Remove parentheses.

y = 3 log ((0) + 5) + 2

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

Step 3.2.3

Simplify each term.

Tap for more steps...

Step 3.2.3.1

Add

0

$0$ and

5

$5$ .

y = 3 log (5) + 2

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

Step 3.2.3.2

Simplify

3 log (5)

$3 \log (5)$ by moving

3

$3$ inside the logarithm.

y = log (5^{3}) + 2

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

Step 3.2.3.3

Raise

5

$5$ to the power of

3

$3$ .

y = log (125) + 2

$y = \log (125) + 2$

y = log (125) + 2

$y = \log (125) + 2$

y = log (125) + 2

$y = \log (125) + 2$

Step 3.3

y-intercept(s) in point form.

y-intercept(s):

(0, log (125) + 2)

$(0, \log (125) + 2)$

y-intercept(s):

(0, log (125) + 2)

$(0, \log (125) + 2)$

Step 4

List the intersections.

x-intercept(s):

(\frac{1}{10^{\frac{2}{3}}} - 5, 0)

$(\frac{1}{10^{\frac{2}{3}}} - 5, 0)$

y-intercept(s):

(0, log (125) + 2)

$(0, \log (125) + 2)$

Step 5