Solve for x log base 5 of 125- log base 5 of square root of 5 = log base 5 of x

$\log_{5} (125) - \log_{5} (\sqrt{5}) = \log_{5} (x)$

Step 1

Rewrite the equation as $\log_{5} (x) = \log_{5} (125) - \log_{5} (\sqrt{5})$ .

$\log_{5} (x) = \log_{5} (125) - \log_{5} (\sqrt{5})$

Step 2

Simplify the right side.

Simplify $\log_{5} (125) - \log_{5} (\sqrt{5})$ .

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

$\log_{5} (x) = \log_{5} (\frac{125}{\sqrt{5}})$

Multiply $\frac{125}{\sqrt{5}}$ by $\frac{\sqrt{5}}{\sqrt{5}}$ .

$\log_{5} (x) = \log_{5} (\frac{125}{\sqrt{5}} \cdot \frac{\sqrt{5}}{\sqrt{5}})$

Combine and simplify the denominator.

Multiply $\frac{125}{\sqrt{5}}$ by $\frac{\sqrt{5}}{\sqrt{5}}$ .

$\log_{5} (x) = \log_{5} (\frac{125 \sqrt{5}}{\sqrt{5} \sqrt{5}})$

Raise $\sqrt{5}$ to the power of $1$ .

$\log_{5} (x) = \log_{5} (\frac{125 \sqrt{5}}{{\sqrt{5}}^{1} \sqrt{5}})$

Raise $\sqrt{5}$ to the power of $1$ .

$\log_{5} (x) = \log_{5} (\frac{125 \sqrt{5}}{{\sqrt{5}}^{1} {\sqrt{5}}^{1}})$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\log_{5} (x) = \log_{5} (\frac{125 \sqrt{5}}{{\sqrt{5}}^{1 + 1}})$

Add $1$ and $1$ .

$\log_{5} (x) = \log_{5} (\frac{125 \sqrt{5}}{{\sqrt{5}}^{2}})$

Rewrite ${\sqrt{5}}^{2}$ as $5$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5}$ as $5^{\frac{1}{2}}$ .

$\log_{5} (x) = \log_{5} (\frac{125 \sqrt{5}}{{(5^{\frac{1}{2}})}^{2}})$

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\log_{5} (x) = \log_{5} (\frac{125 \sqrt{5}}{5^{\frac{1}{2} \cdot 2}})$

Combine $\frac{1}{2}$ and $2$ .

$\log_{5} (x) = \log_{5} (\frac{125 \sqrt{5}}{5^{\frac{2}{2}}})$

Cancel the common factor of $2$ .

Cancel the common factor.

$\log_{5} (x) = \log_{5} (\frac{125 \sqrt{5}}{5^{\frac{2}{2}}})$

Rewrite the expression.

$\log_{5} (x) = \log_{5} (\frac{125 \sqrt{5}}{5^{1}})$

Evaluate the exponent.

$\log_{5} (x) = \log_{5} (\frac{125 \sqrt{5}}{5})$

Cancel the common factor of $125$ and $5$ .

Factor $5$ out of $125 \sqrt{5}$ .

$\log_{5} (x) = \log_{5} (\frac{5 (25 \sqrt{5})}{5})$

Cancel the common factors.

Factor $5$ out of $5$ .

$\log_{5} (x) = \log_{5} (\frac{5 (25 \sqrt{5})}{5 (1)})$

Cancel the common factor.

$\log_{5} (x) = \log_{5} (\frac{5 (25 \sqrt{5})}{5 \cdot 1})$

Rewrite the expression.

$\log_{5} (x) = \log_{5} (\frac{25 \sqrt{5}}{1})$

Divide $25 \sqrt{5}$ by $1$ .

$\log_{5} (x) = \log_{5} (25 \sqrt{5})$

Step 3

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

$x = 25 \sqrt{5}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$x = 25 \sqrt{5}$

Decimal Form:

$x = 55.90169943 \dots$

Solve for x log base 5 of 125- log base 5 of square root of 5 = log base 5 of x

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