Finite Math Examples

$\frac{\log (x - 10)}{\log_{1 x - 100} (25)}$

Step 1

Set the denominator in

$\frac{\log (x - 10)}{\log_{1 x - 100} (25)}$ equal to

$0$ to find where the expression is undefined.

$\log_{1 x - 100} (25) = 0$

Step 2

Solve for

$x$ .

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Step 2.1

Rewrite

$\log_{1 x - 100} (25) = 0$ 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

$b^{y} = x$ .

${(1 x - 100)}^{0} = 25$

Step 2.2

Solve for

$x$ .

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Step 2.2.1

Simplify

${(1 x - 100)}^{0}$ .

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Step 2.2.1.1

Multiply

$x$ by

$1$ .

${(x - 100)}^{0} = 25$

Step 2.2.1.2

Anything raised to

$0$ is

$1$ .

$1 = 25$

Step 2.2.2

Since

$1 \neq 25$ , there are no solutions.

No solution

Step 3

Set the argument in

$\log (x - 10)$ less than or equal to

$0$ to find where the expression is undefined.

$x - 10 \leq 0$

Step 4

Add

$10$ to both sides of the inequality.

$x \leq 10$

Step 5

Set the base in

$\log_{1 x - 100} (25)$ less than or equal to

$0$ to find where the expression is undefined.

$1 x - 100 \leq 0$

Step 6

Solve for

$x$ .

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Step 6.1

Multiply

$x$ by

$1$ .

$x - 100 \leq 0$

Step 6.2

Add

$100$ to both sides of the inequality.

$x \leq 100$

Step 7

Set the base in

$\log_{1 x - 100} (25)$ equal to

$1$ to find where the expression is undefined.

$1 x - 100 = 1$

Step 8

Solve for

$x$ .

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Step 8.1

Multiply

$x$ by

$1$ .

$x - 100 = 1$

Step 8.2

Move all terms not containing

$x$ to the right side of the equation.

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Step 8.2.1

Add

$100$ to both sides of the equation.

$x = 1 + 100$

Step 8.2.2

Add

$1$ and

$100$ .

$x = 101$

Step 9

The equation is undefined where the denominator equals

$0$ , the argument of a square root is less than

$0$ , or the argument of a logarithm is less than or equal to

$0$ .

$x \leq 100, x = 101$

$(- \infty, 100] \cup [101, 101]$

Step 10