Algebra Examples

$f (x) = \log_{\frac{1}{4}} (x - 1) + 1$

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 $x$ .

$0 = \log_{\frac{1}{4}} (x - 1) + 1$

Step 1.2

Solve the equation.

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Step 1.2.1

Rewrite the equation as $\log_{\frac{1}{4}} (x - 1) + 1 = 0$ .

$\log_{\frac{1}{4}} (x - 1) + 1 = 0$

Step 1.2.2

Subtract $1$ from both sides of the equation.

$\log_{\frac{1}{4}} (x - 1) = - 1$

Step 1.2.3

Rewrite $\log_{\frac{1}{4}} (x - 1) = - 1$ 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 $b^{y} = x$ .

${(\frac{1}{4})}^{- 1} = x - 1$

Step 1.2.4

Solve for $x$ .

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Step 1.2.4.1

Rewrite the equation as $x - 1 = {(\frac{1}{4})}^{- 1}$ .

$x - 1 = {(\frac{1}{4})}^{- 1}$

Step 1.2.4.2

Change the sign of the exponent by rewriting the base as its reciprocal.

$x - 1 = 4$

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 $1$ to both sides of the equation.

$x = 4 + 1$

Step 1.2.4.3.2

Add $4$ and $1$ .

$x = 5$

Step 1.3

x-intercept(s) in point form.

x-intercept(s): $(5, 0)$

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 $y$ .

$y = \log_{\frac{1}{4}} ((0) - 1) + 1$

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.

$y = \log_{\frac{1}{4}} ((0) - 1) + 1$

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): $(5, 0)$

y-intercept(s): $None$

Step 4