Trigonometry Examples

$5 \log (x + 1) + 10 = 10 \log (t + 1) + 5$

Step 1

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

$5 \log (x + 1) - 10 \log (t + 1) = - 10 + 5$

Step 2

Add $- 10$ and $5$ .

$5 \log (x + 1) - 10 \log (t + 1) = - 5$

Step 3

Simplify the left side.

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Step 3.1

Simplify $5 \log (x + 1) - 10 \log (t + 1)$ .

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Step 3.1.1

Simplify each term.

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Step 3.1.1.1

Simplify $5 \log (x + 1)$ by moving $5$ inside the logarithm.

$\log ({(x + 1)}^{5}) - 10 \log (t + 1) = - 5$

Step 3.1.1.2

Simplify $- 10 \log (t + 1)$ by moving $10$ inside the logarithm.

$\log ({(x + 1)}^{5}) - \log ({(t + 1)}^{10}) = - 5$

Step 3.1.2

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

$\log (\frac{{(x + 1)}^{5}}{{(t + 1)}^{10}}) = - 5$

Step 4

Rewrite $\log (\frac{{(x + 1)}^{5}}{{(t + 1)}^{10}}) = - 5$ 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 $b^{y} = x$ .

$10^{- 5} = \frac{{(x + 1)}^{5}}{{(t + 1)}^{10}}$

Step 5

Solve for $x$ .

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Step 5.1

Rewrite the equation as $\frac{{(x + 1)}^{5}}{{(t + 1)}^{10}} = 10^{- 5}$ .

$\frac{{(x + 1)}^{5}}{{(t + 1)}^{10}} = 10^{- 5}$

Step 5.2

Multiply both sides by ${(t + 1)}^{10}$ .

$\frac{{(x + 1)}^{5}}{{(t + 1)}^{10}} {(t + 1)}^{10} = 10^{- 5} {(t + 1)}^{10}$

Step 5.3

Simplify.

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Step 5.3.1

Simplify the left side.

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Step 5.3.1.1

Cancel the common factor of ${(t + 1)}^{10}$ .

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Step 5.3.1.1.1

Cancel the common factor.

$\frac{{(x + 1)}^{5}}{{(t + 1)}^{10}} {(t + 1)}^{10} = 10^{- 5} {(t + 1)}^{10}$

Step 5.3.1.1.2

Rewrite the expression.

${(x + 1)}^{5} = 10^{- 5} {(t + 1)}^{10}$

Step 5.3.2

Simplify the right side.

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Step 5.3.2.1

Simplify $10^{- 5} {(t + 1)}^{10}$ .

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Step 5.3.2.1.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

${(x + 1)}^{5} = \frac{1}{10^{5}} {(t + 1)}^{10}$

Step 5.3.2.1.2

Raise $10$ to the power of $5$ .

${(x + 1)}^{5} = \frac{1}{100000} {(t + 1)}^{10}$

Step 5.3.2.1.3

Combine $\frac{1}{100000}$ and ${(t + 1)}^{10}$ .

${(x + 1)}^{5} = \frac{{(t + 1)}^{10}}{100000}$

Step 5.4

Solve for $x$ .

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Step 5.4.1

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x + 1 = \sqrt[5]{\frac{{(t + 1)}^{10}}{100000}}$

Step 5.4.2

Simplify $\sqrt[5]{\frac{{(t + 1)}^{10}}{100000}}$ .

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Step 5.4.2.1

Rewrite ${(t + 1)}^{10}$ as ${({(t + 1)}^{2})}^{5}$ .

$x + 1 = \sqrt[5]{\frac{{({(t + 1)}^{2})}^{5}}{100000}}$

Step 5.4.2.2

Rewrite $100000$ as $10^{5}$ .

$x + 1 = \sqrt[5]{\frac{{({(t + 1)}^{2})}^{5}}{10^{5}}}$

Step 5.4.2.3

Rewrite $\frac{{({(t + 1)}^{2})}^{5}}{10^{5}}$ as ${(\frac{{(t + 1)}^{2}}{10})}^{5}$ .

$x + 1 = \sqrt[5]{{(\frac{{(t + 1)}^{2}}{10})}^{5}}$

Step 5.4.2.4

Pull terms out from under the radical, assuming real numbers.

$x + 1 = \frac{{(t + 1)}^{2}}{10}$

Step 5.4.3

Subtract $1$ from both sides of the equation.

$x = \frac{{(t + 1)}^{2}}{10} - 1$

Step 5.4.4

Simplify $\frac{{(t + 1)}^{2}}{10} - 1$ .

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Step 5.4.4.1

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{10}{10}$ .

$x = \frac{{(t + 1)}^{2}}{10} - 1 \cdot \frac{10}{10}$

Step 5.4.4.2

Simplify terms.

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Step 5.4.4.2.1

Combine $- 1$ and $\frac{10}{10}$ .

$x = \frac{{(t + 1)}^{2}}{10} + \frac{- 1 \cdot 10}{10}$

Step 5.4.4.2.2

Combine the numerators over the common denominator.

$x = \frac{{(t + 1)}^{2} - 1 \cdot 10}{10}$

Step 5.4.4.3

Simplify the numerator.

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Step 5.4.4.3.1

Rewrite ${(t + 1)}^{2}$ as $(t + 1) (t + 1)$ .

$x = \frac{(t + 1) (t + 1) - 1 \cdot 10}{10}$

Step 5.4.4.3.2

Expand $(t + 1) (t + 1)$ using the FOIL Method.

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Step 5.4.4.3.2.1

Apply the distributive property.

$x = \frac{t (t + 1) + 1 (t + 1) - 1 \cdot 10}{10}$

Step 5.4.4.3.2.2

Apply the distributive property.

$x = \frac{t \cdot t + t \cdot 1 + 1 (t + 1) - 1 \cdot 10}{10}$

Step 5.4.4.3.2.3

Apply the distributive property.

$x = \frac{t \cdot t + t \cdot 1 + 1 t + 1 \cdot 1 - 1 \cdot 10}{10}$

Step 5.4.4.3.3

Simplify and combine like terms.

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Step 5.4.4.3.3.1

Simplify each term.

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Step 5.4.4.3.3.1.1

Multiply $t$ by $t$ .

$x = \frac{t^{2} + t \cdot 1 + 1 t + 1 \cdot 1 - 1 \cdot 10}{10}$

Step 5.4.4.3.3.1.2

Multiply $t$ by $1$ .

$x = \frac{t^{2} + t + 1 t + 1 \cdot 1 - 1 \cdot 10}{10}$

Step 5.4.4.3.3.1.3

Multiply $t$ by $1$ .

$x = \frac{t^{2} + t + t + 1 \cdot 1 - 1 \cdot 10}{10}$

Step 5.4.4.3.3.1.4

Multiply $1$ by $1$ .

$x = \frac{t^{2} + t + t + 1 - 1 \cdot 10}{10}$

Step 5.4.4.3.3.2

Add $t$ and $t$ .

$x = \frac{t^{2} + 2 t + 1 - 1 \cdot 10}{10}$

Step 5.4.4.3.4

Multiply $- 1$ by $10$ .

$x = \frac{t^{2} + 2 t + 1 - 10}{10}$

Step 5.4.4.3.5

Subtract $10$ from $1$ .

$x = \frac{t^{2} + 2 t - 9}{10}$