Algebra Examples

$y = \log (7 x + 3) - 2$

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 $x$ .

$0 = \log (7 x + 3) - 2$

Step 1.2

Solve the equation.

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Step 1.2.1

Rewrite the equation as $\log (7 x + 3) - 2 = 0$ .

$\log (7 x + 3) - 2 = 0$

Step 1.2.2

Add $2$ to both sides of the equation.

$\log (7 x + 3) = 2$

Step 1.2.3

Rewrite $\log (7 x + 3) = 2$ 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 $b^{y} = x$ .

$10^{2} = 7 x + 3$

Step 1.2.4

Solve for $x$ .

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Step 1.2.4.1

Rewrite the equation as $7 x + 3 = 10^{2}$ .

$7 x + 3 = 10^{2}$

Step 1.2.4.2

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

$7 x + 3 = 100$

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 $3$ from both sides of the equation.

$7 x = 100 - 3$

Step 1.2.4.3.2

Subtract $3$ from $100$ .

$7 x = 97$

Step 1.2.4.4

Divide each term in $7 x = 97$ by $7$ and simplify.

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Step 1.2.4.4.1

Divide each term in $7 x = 97$ by $7$ .

$\frac{7 x}{7} = \frac{97}{7}$

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 $7$ .

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Step 1.2.4.4.2.1.1

Cancel the common factor.

$\frac{7 x}{7} = \frac{97}{7}$

Step 1.2.4.4.2.1.2

Divide $x$ by $1$ .

$x = \frac{97}{7}$

Step 1.3

x-intercept(s) in point form.

x-intercept(s): $(\frac{97}{7}, 0)$

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 $y$ .

$y = \log (7 (0) + 3) - 2$

Step 2.2

Solve the equation.

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Step 2.2.1

Remove parentheses.

$y = \log (7 (0) + 3) - 2$

Step 2.2.2

Simplify each term.

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Step 2.2.2.1

Multiply $7$ by $0$ .

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

Step 2.2.2.2

Add $0$ and $3$ .

$y = \log (3) - 2$

Step 2.3

y-intercept(s) in point form.

y-intercept(s): $(0, \log (3) - 2)$

Step 3

List the intersections.

x-intercept(s): $(\frac{97}{7}, 0)$

y-intercept(s): $(0, \log (3) - 2)$

Step 4