Precalculus Examples

$y = \log_{5} (x - 2) + 3$

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 $x$ .

$0 = \log_{5} (x - 2) + 3$

Step 1.2

Solve the equation.

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Step 1.2.1

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

$\log_{5} (x - 2) + 3 = 0$

Step 1.2.2

Subtract $3$ from both sides of the equation.

$\log_{5} (x - 2) = - 3$

Step 1.2.3

Rewrite $\log_{5} (x - 2) = - 3$ 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 $b^{y} = x$ .

$5^{- 3} = x - 2$

Step 1.2.4

Solve for $x$ .

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Step 1.2.4.1

Rewrite the equation as $x - 2 = 5^{- 3}$ .

$x - 2 = 5^{- 3}$

Step 1.2.4.2

Simplify $5^{- 3}$ .

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Step 1.2.4.2.1

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

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

Step 1.2.4.2.2

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

$x - 2 = \frac{1}{125}$

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

Add $2$ to both sides of the equation.

$x = \frac{1}{125} + 2$

Step 1.2.4.3.2

To write $2$ as a fraction with a common denominator, multiply by $\frac{125}{125}$ .

$x = \frac{1}{125} + 2 \cdot \frac{125}{125}$

Step 1.2.4.3.3

Combine $2$ and $\frac{125}{125}$ .

$x = \frac{1}{125} + \frac{2 \cdot 125}{125}$

Step 1.2.4.3.4

Combine the numerators over the common denominator.

$x = \frac{1 + 2 \cdot 125}{125}$

Step 1.2.4.3.5

Simplify the numerator.

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Step 1.2.4.3.5.1

Multiply $2$ by $125$ .

$x = \frac{1 + 250}{125}$

Step 1.2.4.3.5.2

Add $1$ and $250$ .

$x = \frac{251}{125}$

Step 1.3

x-intercept(s) in point form.

x-intercept(s): $(\frac{251}{125}, 0)$

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 $y$ .

$y = \log_{5} ((0) - 2) + 3$

Step 2.2

Solve the equation.

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Step 2.2.1

The logarithm of a negative number is undefined.

$y = Undefined$

Step 2.2.2

Remove parentheses.

$y = \log_{5} ((0) - 2) + 3$

Step 2.2.3

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): $(\frac{251}{125}, 0)$

y-intercept(s): $None$

Step 4