Solve for p 1/3* log base 5 of 64 = log base 5 of 8+ log base 5 of p

$\frac{1}{3} \cdot \log_{5} (64) = \log_{5} (8) + \log_{5} (p)$

Step 1

Rewrite the equation as $\log_{5} (8) + \log_{5} (p) = \frac{1}{3} \cdot \log_{5} (64)$ .

$\log_{5} (8) + \log_{5} (p) = \frac{1}{3} \cdot \log_{5} (64)$

Step 2

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

$\log_{5} (8 p) = \frac{1}{3} \cdot \log_{5} (64)$

Step 3

Simplify the right side.

Combine $\frac{1}{3}$ and $\log_{5} (64)$ .

$\log_{5} (8 p) = \frac{\log_{5} (64)}{3}$

Rewrite $\log_{5} (64)$ as $\log_{5} (2^{6})$ .

$\log_{5} (8 p) = \frac{\log_{5} (2^{6})}{3}$

Expand $\log_{5} (2^{6})$ by moving $6$ outside the logarithm.

$\log_{5} (8 p) = \frac{6 \log_{5} (2)}{3}$

Cancel the common factor of $6$ and $3$ .

Factor $3$ out of $6 \log_{5} (2)$ .

$\log_{5} (8 p) = \frac{3 (2 \log_{5} (2))}{3}$

Cancel the common factors.

Factor $3$ out of $3$ .

$\log_{5} (8 p) = \frac{3 (2 \log_{5} (2))}{3 (1)}$

Cancel the common factor.

$\log_{5} (8 p) = \frac{3 (2 \log_{5} (2))}{3 \cdot 1}$

Rewrite the expression.

$\log_{5} (8 p) = \frac{2 \log_{5} (2)}{1}$

Divide $2 \log_{5} (2)$ by $1$ .

$\log_{5} (8 p) = 2 \log_{5} (2)$

Step 4

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

$\log_{5} (8 p) = \log_{5} (2^{2})$

Step 5

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

$8 p = 2^{2}$

Step 6

Divide each term in $8 p = 2^{2}$ by $8$ and simplify.

Divide each term in $8 p = 2^{2}$ by $8$ .

$\frac{8 p}{8} = \frac{2^{2}}{8}$

Simplify the left side.

Cancel the common factor of $8$ .

Cancel the common factor.

$\frac{8 p}{8} = \frac{2^{2}}{8}$

Divide $p$ by $1$ .

$p = \frac{2^{2}}{8}$

Simplify the right side.

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

$p = \frac{4}{8}$

Cancel the common factor of $4$ and $8$ .

Factor $4$ out of $4$ .

$p = \frac{4 (1)}{8}$

Cancel the common factors.

Factor $4$ out of $8$ .

$p = \frac{4 \cdot 1}{4 \cdot 2}$

Cancel the common factor.

$p = \frac{4 \cdot 1}{4 \cdot 2}$

Rewrite the expression.

$p = \frac{1}{2}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$p = \frac{1}{2}$

Decimal Form:

$p = 0.5$

Solve for p 1/3* log base 5 of 64 = log base 5 of 8+ log base 5 of p

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