Algebra Examples

$f (x) = 2 \log (5) (x + 4) - 6$

Step 1

Find the x-intercepts.

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Step 1.1

To find the x-intercept(s), substitute in $0$ for $y$ and solve for $x$ .

$0 = 2 \log (5) (x + 4) - 6$

Step 1.2

Solve the equation.

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Step 1.2.1

Rewrite the equation as $2 \log (5) (x + 4) - 6 = 0$ .

$2 \log (5) (x + 4) - 6 = 0$

Step 1.2.2

Simplify $2 \log (5) (x + 4) - 6$ .

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Step 1.2.2.1

Simplify each term.

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Step 1.2.2.1.1

Simplify $2 \log (5)$ by moving $2$ inside the logarithm.

$\log (5^{2}) (x + 4) - 6 = 0$

Step 1.2.2.1.2

Raise $5$ to the power of $2$ .

$\log (25) (x + 4) - 6 = 0$

Step 1.2.2.1.3

Apply the distributive property.

$\log (25) x + \log (25) \cdot 4 - 6 = 0$

Step 1.2.2.1.4

Multiply $\log (25) \cdot 4$ .

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Step 1.2.2.1.4.1

Reorder $\log (25)$ and $4$ .

$\log (25) x + 4 \cdot \log (25) - 6 = 0$

Step 1.2.2.1.4.2

Simplify $4 \cdot \log (25)$ by moving $4$ inside the logarithm.

$\log (25) x + \log (25^{4}) - 6 = 0$

Step 1.2.2.1.5

Raise $25$ to the power of $4$ .

$\log (25) x + \log (390625) - 6 = 0$

Step 1.2.2.2

Reorder factors in $\log (25) x + \log (390625) - 6$ .

$x \log (25) + \log (390625) - 6 = 0$

Step 1.2.3

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

$x \log (25) + \log (390625) = 6 + 0$

Step 1.2.4

Add $6$ and $0$ .

$x \log (25) + \log (390625) = 6$

Step 1.2.5

Subtract $\log (390625)$ from both sides of the equation.

$x \log (25) = 6 - \log (390625)$

Step 1.2.6

Divide each term in $x \log (25) = 6 - \log (390625)$ by $\log (25)$ and simplify.

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Step 1.2.6.1

Divide each term in $x \log (25) = 6 - \log (390625)$ by $\log (25)$ .

$\frac{x \log (25)}{\log (25)} = \frac{6}{\log (25)} + \frac{- \log (390625)}{\log (25)}$

Step 1.2.6.2

Simplify the left side.

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Step 1.2.6.2.1

Cancel the common factor of $\log (25)$ .

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Step 1.2.6.2.1.1

Cancel the common factor.

$\frac{x \log (25)}{\log (25)} = \frac{6}{\log (25)} + \frac{- \log (390625)}{\log (25)}$

Step 1.2.6.2.1.2

Divide $x$ by $1$ .

$x = \frac{6}{\log (25)} + \frac{- \log (390625)}{\log (25)}$

Step 1.2.6.3

Simplify the right side.

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Step 1.2.6.3.1

Move the negative in front of the fraction.

$x = \frac{6}{\log (25)} - \frac{\log (390625)}{\log (25)}$

Step 1.3

x-intercept(s) in point form.

x-intercept(s): $(\frac{6}{\log (25)} - \frac{\log (390625)}{\log (25)}, 0)$

Step 2

Find the y-intercepts.

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Step 2.1

To find the y-intercept(s), substitute in $0$ for $x$ and solve for $y$ .

$y = 2 \log (5) ((0) + 4) - 6$

Step 2.2

Solve the equation.

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Step 2.2.1

Remove parentheses.

$y = 2 \log (5) (0 + 4) - 6$

Step 2.2.2

Remove parentheses.

$y = 2 \log (5) ((0) + 4) - 6$

Step 2.2.3

Simplify each term.

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Step 2.2.3.1

Simplify $2 \log (5)$ by moving $2$ inside the logarithm.

$y = \log (5^{2}) ((0) + 4) - 6$

Step 2.2.3.2

Raise $5$ to the power of $2$ .

$y = \log (25) ((0) + 4) - 6$

Step 2.2.3.3

Add $0$ and $4$ .

$y = \log (25) \cdot 4 - 6$

Step 2.2.3.4

Multiply $\log (25) \cdot 4$ .

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Step 2.2.3.4.1

Reorder $\log (25)$ and $4$ .

$y = 4 \cdot \log (25) - 6$

Step 2.2.3.4.2

Simplify $4 \cdot \log (25)$ by moving $4$ inside the logarithm.

$y = \log (25^{4}) - 6$

Step 2.2.3.5

Raise $25$ to the power of $4$ .

$y = \log (390625) - 6$

Step 2.3

y-intercept(s) in point form.

y-intercept(s): $(0, \log (390625) - 6)$

Step 3

List the intersections.

x-intercept(s): $(\frac{6}{\log (25)} - \frac{\log (390625)}{\log (25)}, 0)$

y-intercept(s): $(0, \log (390625) - 6)$

Step 4