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