Algebra Examples

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

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

Step 1

Simplify

\frac{\log_{4} (x)}{\log_{4} (4)}

$\frac{\log_{4} (x)}{\log_{4} (4)}$ .

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Step 1.1

Logarithm base

4

$4$ of

4

$4$ is

1

$1$ .

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

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

Step 1.2

Divide

\log_{4} (x)

$\log_{4} (x)$ by

1

$1$ .

\log_{4} (x - 4) = \log_{4} (x)

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

\log_{4} (x - 4) = \log_{4} (x)

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

Step 2

Move all terms containing

x

$x$ to the left side of the equation.

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Step 2.1

Subtract

\log_{4} (x)

$\log_{4} (x)$ from both sides of the equation.

\log_{4} (x - 4) - \log_{4} (x) = 0

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

Step 2.2

Use the quotient property of logarithms,

\log_{b} (x) - \log_{b} (y) = \log_{b} (\frac{x}{y})

$\log_{b} (x) - \log_{b} (y) = \log_{b} (\frac{x}{y})$ .

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

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

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

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

Step 3

Rewrite

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

$\log_{4} (\frac{x - 4}{x}) = 0$ in exponential form using the definition of a logarithm. If

x

$x$ and

b

$b$ are positive real numbers and

b

$b$

\neq

$\neq$

1

$1$ , then

\log_{b} (x) = y

$\log_{b} (x) = y$ is equivalent to

b^{y} = x

$b^{y} = x$ .

4^{0} = \frac{x - 4}{x}

$4^{0} = \frac{x - 4}{x}$

Step 4

Cross multiply to remove the fraction.

x - 4 = 4^{0} (x)

$x - 4 = 4^{0} (x)$

Step 5

Simplify

4^{0} (x)

$4^{0} (x)$ .

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Step 5.1

Remove parentheses.

x - 4 = 4^{0} (x)

$x - 4 = 4^{0} (x)$

Step 5.2

Anything raised to

0

$0$ is

1

$1$ .

x - 4 = 1 x

$x - 4 = 1 x$

Step 5.3

Multiply

x

$x$ by

1

$1$ .

x - 4 = x

$x - 4 = x$

x - 4 = x

$x - 4 = x$

Step 6

Move all terms containing

x

$x$ to the left side of the equation.

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Step 6.1

Subtract

x

$x$ from both sides of the equation.

x - 4 - x = 0

$x - 4 - x = 0$

Step 6.2

Combine the opposite terms in

x - 4 - x

$x - 4 - x$ .

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Step 6.2.1

Subtract

x

$x$ from

x

$x$ .

0 - 4 = 0

$0 - 4 = 0$

Step 6.2.2

Subtract

4

$4$ from

0

$0$ .

- 4 = 0

$- 4 = 0$

- 4 = 0

$- 4 = 0$

- 4 = 0

$- 4 = 0$

Step 7

Add

4

$4$ to both sides of the equation.

0 = 4

$0 = 4$

Step 8

The equation is never true.

No solution