Algebra Examples

$\log (b x) - \log (b \cdot 5) = \log (b \cdot 2) - \log (b) (x - 3)$

Step 1

Simplify the left side.

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Step 1.1

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

$\log (\frac{b x}{b \cdot 5}) = \log (b \cdot 2) - \log (b) (x - 3)$

Step 1.2

Cancel the common factor of $b$ .

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Step 1.2.1

Cancel the common factor.

$\log (\frac{b x}{b \cdot 5}) = \log (b \cdot 2) - \log (b) (x - 3)$

Step 1.2.2

Rewrite the expression.

$\log (\frac{x}{5}) = \log (b \cdot 2) - \log (b) (x - 3)$

Step 2

Simplify the right side.

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Step 2.1

Simplify each term.

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Step 2.1.1

Move $2$ to the left of $b$ .

$\log (\frac{x}{5}) = \log (2 \cdot b) - \log (b) (x - 3)$

Step 2.1.2

Apply the distributive property.

$\log (\frac{x}{5}) = \log (2 b) - \log (b) x - \log (b) \cdot - 3$

Step 2.1.3

Multiply $- 3$ by $- 1$ .

$\log (\frac{x}{5}) = \log (2 b) - \log (b) x + 3 \log (b)$

Step 2.2

Simplify the expression.

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Step 2.2.1

Reorder factors in $\log (2 b) - \log (b) x + 3 \log (b)$ .

$\log (\frac{x}{5}) = \log (2 b) - x \log (b) + 3 \log (b)$

Step 2.2.2

Reorder $\log (2 b)$ and $- x \log (b)$ .

$\log (\frac{x}{5}) = - x \log (b) + \log (2 b) + 3 \log (b)$

Step 3

Simplify the right side.

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Step 3.1

Simplify $- x \log (b) + \log (2 b) + 3 \log (b)$ .

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Step 3.1.1

Simplify $3 \log (b)$ by moving $3$ inside the logarithm.

$\log (\frac{x}{5}) = - x \log (b) + \log (2 b) + \log (b^{3})$

Step 3.1.2

Use the product property of logarithms, $\log_{b} (x) + \log_{b} (y) = \log_{b} (x y)$ .

$\log (\frac{x}{5}) = - x \log (b) + \log (2 b \cdot b^{3})$

Step 3.1.3

Multiply $b$ by $b^{3}$ by adding the exponents.

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Step 3.1.3.1

Move $b^{3}$ .

$\log (\frac{x}{5}) = - x \log (b) + \log (2 (b^{3} b))$

Step 3.1.3.2

Multiply $b^{3}$ by $b$ .

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Step 3.1.3.2.1

Raise $b$ to the power of $1$ .

$\log (\frac{x}{5}) = - x \log (b) + \log (2 (b^{3} b^{1}))$

Step 3.1.3.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\log (\frac{x}{5}) = - x \log (b) + \log (2 b^{3 + 1})$

Step 3.1.3.3

Add $3$ and $1$ .

$\log (\frac{x}{5}) = - x \log (b) + \log (2 b^{4})$

Step 4

Move all the terms containing a logarithm to the left side of the equation.

$\log (\frac{x}{5}) + x \log (b) - \log (2 b^{4}) = 0$

Step 5

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

$x \log (b) + \log (\frac{\frac{x}{5}}{2 b^{4}}) = 0$

Step 6

Simplify each term.

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Step 6.1

Multiply the numerator by the reciprocal of the denominator.

$x \log (b) + \log (\frac{x}{5} \cdot \frac{1}{2 b^{4}}) = 0$

Step 6.2

Combine.

$x \log (b) + \log (\frac{x \cdot 1}{5 (2 b^{4})}) = 0$

Step 6.3

Multiply $x$ by $1$ .

$x \log (b) + \log (\frac{x}{5 \cdot (2 b^{4})}) = 0$

Step 6.4

Multiply $5$ by $2$ .

$x \log (b) + \log (\frac{x}{10 b^{4}}) = 0$

Step 7

Rewrite $\log (\frac{x}{10 b^{4}}) = - x \log (b)$ 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 $b^{y} = x$ .

$10^{- x \log (b)} = \frac{x}{10 b^{4}}$

Step 8

Solve for $b$ .

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Step 8.1

Take the natural logarithm of both sides of the equation to remove the variable from the exponent.

$\ln (10^{- x \log (b)}) = \ln (\frac{x}{10 b^{4}})$

Step 8.2

Expand $\ln (10^{- x \log (b)})$ by moving $- x \log (b)$ outside the logarithm.

$- x \log (b) \ln (10) = \ln (\frac{x}{10 b^{4}})$