Algebra Examples

$y = \log (3 x + 4) + 2$

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 $x$ .

$0 = \log (3 x + 4) + 2$

Step 1.2

Solve the equation.

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Step 1.2.1

Rewrite the equation as $\log (3 x + 4) + 2 = 0$ .

$\log (3 x + 4) + 2 = 0$

Step 1.2.2

Subtract $2$ from both sides of the equation.

$\log (3 x + 4) = - 2$

Step 1.2.3

Rewrite $\log (3 x + 4) = - 2$ 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 $b^{y} = x$ .

$10^{- 2} = 3 x + 4$

Step 1.2.4

Solve for $x$ .

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Step 1.2.4.1

Rewrite the equation as $3 x + 4 = 10^{- 2}$ .

$3 x + 4 = 10^{- 2}$

Step 1.2.4.2

Simplify $10^{- 2}$ .

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Step 1.2.4.2.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$3 x + 4 = \frac{1}{10^{2}}$

Step 1.2.4.2.2

Raise $10$ to the power of $2$ .

$3 x + 4 = \frac{1}{100}$

Step 1.2.4.3

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

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Step 1.2.4.3.1

Subtract $4$ from both sides of the equation.

$3 x = \frac{1}{100} - 4$

Step 1.2.4.3.2

To write $- 4$ as a fraction with a common denominator, multiply by $\frac{100}{100}$ .

$3 x = \frac{1}{100} - 4 \cdot \frac{100}{100}$

Step 1.2.4.3.3

Combine $- 4$ and $\frac{100}{100}$ .

$3 x = \frac{1}{100} + \frac{- 4 \cdot 100}{100}$

Step 1.2.4.3.4

Combine the numerators over the common denominator.

$3 x = \frac{1 - 4 \cdot 100}{100}$

Step 1.2.4.3.5

Simplify the numerator.

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Step 1.2.4.3.5.1

Multiply $- 4$ by $100$ .

$3 x = \frac{1 - 400}{100}$

Step 1.2.4.3.5.2

Subtract $400$ from $1$ .

$3 x = \frac{- 399}{100}$

Step 1.2.4.3.6

Move the negative in front of the fraction.

$3 x = - \frac{399}{100}$

Step 1.2.4.4

Divide each term in $3 x = - \frac{399}{100}$ by $3$ and simplify.

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Step 1.2.4.4.1

Divide each term in $3 x = - \frac{399}{100}$ by $3$ .

$\frac{3 x}{3} = \frac{- \frac{399}{100}}{3}$

Step 1.2.4.4.2

Simplify the left side.

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Step 1.2.4.4.2.1

Cancel the common factor of $3$ .

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Step 1.2.4.4.2.1.1

Cancel the common factor.

$\frac{3 x}{3} = \frac{- \frac{399}{100}}{3}$

Step 1.2.4.4.2.1.2

Divide $x$ by $1$ .

$x = \frac{- \frac{399}{100}}{3}$

Step 1.2.4.4.3

Simplify the right side.

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Step 1.2.4.4.3.1

Multiply the numerator by the reciprocal of the denominator.

$x = - \frac{399}{100} \cdot \frac{1}{3}$

Step 1.2.4.4.3.2

Cancel the common factor of $3$ .

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Step 1.2.4.4.3.2.1

Move the leading negative in $- \frac{399}{100}$ into the numerator.

$x = \frac{- 399}{100} \cdot \frac{1}{3}$

Step 1.2.4.4.3.2.2

Factor $3$ out of $- 399$ .

$x = \frac{3 (- 133)}{100} \cdot \frac{1}{3}$

Step 1.2.4.4.3.2.3

Cancel the common factor.

$x = \frac{3 \cdot - 133}{100} \cdot \frac{1}{3}$

Step 1.2.4.4.3.2.4

Rewrite the expression.

$x = \frac{- 133}{100}$

Step 1.2.4.4.3.3

Move the negative in front of the fraction.

$x = - \frac{133}{100}$

Step 1.3

x-intercept(s) in point form.

x-intercept(s): $(- \frac{133}{100}, 0)$

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 $y$ .

$y = \log (3 (0) + 4) + 2$

Step 2.2

Solve the equation.

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Step 2.2.1

Remove parentheses.

$y = \log (3 (0) + 4) + 2$

Step 2.2.2

Simplify each term.

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Step 2.2.2.1

Multiply $3$ by $0$ .

$y = \log (0 + 4) + 2$

Step 2.2.2.2

Add $0$ and $4$ .

$y = \log (4) + 2$

Step 2.3

y-intercept(s) in point form.

y-intercept(s): $(0, \log (4) + 2)$

Step 3

List the intersections.

x-intercept(s): $(- \frac{133}{100}, 0)$

y-intercept(s): $(0, \log (4) + 2)$

Step 4