Trigonometry Examples

$\log (x + 1) = 4 \log (2) + \log (3) - 3 \log (5)$

Step 1

Simplify the right side.

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Step 1.1

Simplify $4 \log (2) + \log (3) - 3 \log (5)$ .

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Step 1.1.1

Simplify each term.

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Step 1.1.1.1

Simplify $4 \log (2)$ by moving $4$ inside the logarithm.

$\log (x + 1) = \log (2^{4}) + \log (3) - 3 \log (5)$

Step 1.1.1.2

Raise $2$ to the power of $4$ .

$\log (x + 1) = \log (16) + \log (3) - 3 \log (5)$

Step 1.1.1.3

Simplify $- 3 \log (5)$ by moving $3$ inside the logarithm.

$\log (x + 1) = \log (16) + \log (3) - \log (5^{3})$

Step 1.1.1.4

Raise $5$ to the power of $3$ .

$\log (x + 1) = \log (16) + \log (3) - \log (125)$

Step 1.1.2

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

$\log (x + 1) = \log (16 \cdot 3) - \log (125)$

Step 1.1.3

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

$\log (x + 1) = \log (\frac{16 \cdot 3}{125})$

Step 1.1.4

Multiply $16$ by $3$ .

$\log (x + 1) = \log (\frac{48}{125})$

Step 2

For the equation to be equal, the argument of the logarithms on both sides of the equation must be equal.

$x + 1 = \frac{48}{125}$

Step 3

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

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Step 3.1

Subtract $1$ from both sides of the equation.

$x = \frac{48}{125} - 1$

Step 3.2

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{125}{125}$ .

$x = \frac{48}{125} - 1 \cdot \frac{125}{125}$

Step 3.3

Combine $- 1$ and $\frac{125}{125}$ .

$x = \frac{48}{125} + \frac{- 1 \cdot 125}{125}$

Step 3.4

Combine the numerators over the common denominator.

$x = \frac{48 - 1 \cdot 125}{125}$

Step 3.5

Simplify the numerator.

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Step 3.5.1

Multiply $- 1$ by $125$ .

$x = \frac{48 - 125}{125}$

Step 3.5.2

Subtract $125$ from $48$ .

$x = \frac{- 77}{125}$

Step 3.6

Move the negative in front of the fraction.

$x = - \frac{77}{125}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{77}{125}$

Decimal Form:

$x = - 0.616$