Finite Math Examples

$\log (\log (4 + b)) = \log (3 c - 1)$

Step 1

Subtract $\log (3 c - 1)$ from both sides of the equation.

$\log (\log (4 + b)) - \log (3 c - 1) = 0$

Step 2

Use the quotient property of logarithms, $\log_{b} (x) - \log_{b} (y) = \log_{b} (\frac{x}{y})$ .

$\log (\frac{\log (4 + b)}{3 c - 1}) = 0$

Step 3

Set the denominator in $\frac{\log (4 + b)}{3 c - 1}$ equal to $0$ to find where the expression is undefined.

$3 c - 1 = 0$

Step 4

Solve for $b$ .

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Step 4.1

Add $1$ to both sides of the equation.

$3 c = 1$

Step 4.2

Divide each term in $3 c = 1$ by $3$ and simplify.

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Step 4.2.1

Divide each term in $3 c = 1$ by $3$ .

$\frac{3 c}{3} = \frac{1}{3}$

Step 4.2.2

Simplify the left side.

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Step 4.2.2.1

Cancel the common factor of $3$ .

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Step 4.2.2.1.1

Cancel the common factor.

$\frac{3 c}{3} = \frac{1}{3}$

Step 4.2.2.1.2

Divide $c$ by $1$ .

$c = \frac{1}{3}$

Step 5

Set the argument in $\log (4 + b)$ less than or equal to $0$ to find where the expression is undefined.

$4 + b \leq 0$

Step 6

Subtract $4$ from both sides of the inequality.

$b \leq - 4$

Step 7

Set the argument in $\log (\frac{\log (4 + b)}{3 c - 1})$ less than or equal to $0$ to find where the expression is undefined.

$\frac{\log (4 + b)}{3 c - 1} \leq 0$

Step 8

Solve for $b$ .

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Step 8.1

Multiply both sides by $3 c - 1$ .

$\frac{\log (4 + b)}{3 c - 1} (3 c - 1) \leq 0 (3 c - 1)$

Step 8.2

Simplify.

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Step 8.2.1

Simplify the left side.

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Step 8.2.1.1

Cancel the common factor of $3 c - 1$ .

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Step 8.2.1.1.1

Cancel the common factor.

$\frac{\log (4 + b)}{3 c - 1} (3 c - 1) \leq 0 (3 c - 1)$

Step 8.2.1.1.2

Rewrite the expression.

$\log (4 + b) \leq 0 (3 c - 1)$

Step 8.2.2

Simplify the right side.

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Step 8.2.2.1

Multiply $0$ by $3 c - 1$ .

$\log (4 + b) \leq 0$

Step 8.3

Solve for $b$ .

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Step 8.3.1

Convert the inequality to an equality.

$\log (4 + b) = 0$

Step 8.3.2

Solve the equation.

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Step 8.3.2.1

Rewrite $\log (4 + b) = 0$ 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 $b^{y} = x$ .

$10^{0} = 4 + b$

Step 8.3.2.2

Solve for $b$ .

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Step 8.3.2.2.1

Rewrite the equation as $4 + b = 10^{0}$ .

$4 + b = 10^{0}$

Step 8.3.2.2.2

Anything raised to $0$ is $1$ .

$4 + b = 1$

Step 8.3.2.2.3

Move all terms not containing $b$ to the right side of the equation.

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Step 8.3.2.2.3.1

Subtract $4$ from both sides of the equation.

$b = 1 - 4$

Step 8.3.2.2.3.2

Subtract $4$ from $1$ .

$b = - 3$

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 $0$ .

$b \leq - 4, b = - 3, b = \frac{1}{3}$

$(- \infty, - 4] \cup [- 3, - 3] \cup [\frac{1}{3}, \frac{1}{3}]$

Step 10