Precalculus Examples

$\log ({(8)}^{2 x^{5}}) + \log ({(8)}^{x}) - 3 \log ({(8)}^{x^{2}})$

Step 1

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

$\log ({(8)}^{2 x^{5}} {(8)}^{x}) - 3 \log ({(8)}^{x^{2}})$

Step 2

Simplify each term.

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Step 2.1

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

$\log (8^{2 x^{5} + x}) - 3 \log ({(8)}^{x^{2}})$

Step 2.2

Simplify $- 3 \log (8^{x^{2}})$ by moving $3$ inside the logarithm.

$\log (8^{2 x^{5} + x}) - \log ({(8^{x^{2}})}^{3})$

Step 2.3

Multiply the exponents in ${(8^{x^{2}})}^{3}$ .

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Step 2.3.1

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

$\log (8^{2 x^{5} + x}) - \log (8^{x^{2} \cdot 3})$

Step 2.3.2

Move $3$ to the left of $x^{2}$ .

$\log (8^{2 x^{5} + x}) - \log (8^{3 x^{2}})$

Step 3

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

$\log (\frac{8^{2 x^{5} + x}}{8^{3 x^{2}}})$

Step 4

Cancel the common factor of $8^{2 x^{5} + x}$ and $8^{3 x^{2}}$ .

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Step 4.1

Factor $8^{3 x^{2}}$ out of $8^{2 x^{5} + x}$ .

$\log (\frac{8^{3 x^{2}} \cdot 8^{2 x^{5} + x - 3 x^{2}}}{8^{3 x^{2}}})$

Step 4.2

Cancel the common factors.

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Step 4.2.1

Multiply by $1$ .

$\log (\frac{8^{3 x^{2}} \cdot 8^{2 x^{5} + x - 3 x^{2}}}{8^{3 x^{2}} \cdot 1})$

Step 4.2.2

Cancel the common factor.

$\log (\frac{8^{3 x^{2}} \cdot 8^{2 x^{5} + x - 3 x^{2}}}{8^{3 x^{2}} \cdot 1})$

Step 4.2.3

Rewrite the expression.

$\log (\frac{8^{2 x^{5} + x - 3 x^{2}}}{1})$

Step 4.2.4

Divide $8^{2 x^{5} + x - 3 x^{2}}$ by $1$ .

$\log (8^{2 x^{5} + x - 3 x^{2}})$