Trigonometry Examples

$\log_{7} (49 x^{3} \sqrt[5]{y^{2}}) = 7 + 3 \log_{7} (x) + \frac{2}{5} \cdot \log_{7} (y)$

Step 1

Solve the equation for $y$ .

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Step 1.1

Simplify both sides of the equation.

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Step 1.1.1

Simplify each term.

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Step 1.1.1.1

Simplify $3 \log_{7} (x)$ by moving $3$ inside the logarithm.

$\log_{7} (49 x^{3} \sqrt[5]{y^{2}}) = 7 + \log_{7} (x^{3}) + \frac{2}{5} \cdot \log_{7} (y)$

Step 1.1.1.2

Simplify $\frac{2}{5} \log_{7} (y)$ by moving $\frac{2}{5}$ inside the logarithm.

$\log_{7} (49 x^{3} \sqrt[5]{y^{2}}) = 7 + \log_{7} (x^{3}) + \log_{7} (y^{\frac{2}{5}})$

Step 1.1.2

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

$\log_{7} (49 x^{3} \sqrt[5]{y^{2}}) = 7 + \log_{7} (x^{3} y^{\frac{2}{5}})$

Step 1.2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[5]{y^{2}}$ as $y^{\frac{2}{5}}$ .

$\log_{7} (49 x^{3} y^{\frac{2}{5}}) = 7 + \log_{7} (x^{3} y^{\frac{2}{5}})$

Step 1.3

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

$\log_{7} (49 x^{3} y^{\frac{2}{5}}) - \log_{7} (x^{3} y^{\frac{2}{5}}) = 7$

Step 1.4

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

$\log_{7} (\frac{49 x^{3} y^{\frac{2}{5}}}{x^{3} y^{\frac{2}{5}}}) = 7$

Step 1.5

Cancel the common factor.

$\log_{7} (\frac{49 x^{3} y^{\frac{2}{5}}}{x^{3} y^{\frac{2}{5}}}) = 7$

Step 1.6

Rewrite the expression.

$\log_{7} (\frac{49 y^{\frac{2}{5}}}{y^{\frac{2}{5}}}) = 7$

Step 1.7

Cancel the common factor.

$\log_{7} (\frac{49 y^{\frac{2}{5}}}{y^{\frac{2}{5}}}) = 7$

Step 1.8

Simplify the expression.

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Step 1.8.1

Divide $49$ by $1$ .

$\log_{7} (49) = 7$

Step 1.8.2

Logarithm base $7$ of $49$ is $2$ .

$2 = 7$

Step 1.9

Since $2 \neq 7$ , there are no solutions.

No solution

Step 2

A linear equation is an equation of a straight line, which means that the degree of a linear equation must be $0$ or $1$ for each of its variables. In this case, there are no variables in the equation.

Linear