Solve for x log base 5 of 3x+3 log base 5 of 3=5

$\log_{5} (3 x) + 3 \log_{5} (3) = 5$

Step 1

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

$\log_{5} (3 x) + 3 \log_{5} (3) = 5$

Step 2

Simplify the left side.

Simplify $\log_{5} (3 x) + 3 \log_{5} (3)$ .

Simplify each term.

Simplify $3 \log_{5} (3)$ by moving $3$ inside the logarithm.

$\log_{5} (3 x) + \log_{5} (3^{3}) = 5$

Raise $3$ to the power of $3$ .

$\log_{5} (3 x) + \log_{5} (27) = 5$

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

$\log_{5} (3 x \cdot 27) = 5$

Multiply $27$ by $3$ .

$\log_{5} (81 x) = 5$

Step 3

Rewrite $\log_{5} (81 x) = 5$ 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 $b^{y} = x$ .

$5^{5} = 81 x$

Step 4

Solve for $x$ .

Rewrite the equation as $81 x = 5^{5}$ .

$81 x = 5^{5}$

Divide each term in $81 x = 5^{5}$ by $81$ and simplify.

Divide each term in $81 x = 5^{5}$ by $81$ .

$\frac{81 x}{81} = \frac{5^{5}}{81}$

Simplify the left side.

Cancel the common factor of $81$ .

Cancel the common factor.

$\frac{81 x}{81} = \frac{5^{5}}{81}$

Divide $x$ by $1$ .

$x = \frac{5^{5}}{81}$

Simplify the right side.

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

$x = \frac{3125}{81}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = \frac{3125}{81}$

Decimal Form:

$x = 38.58024691 \dots$

Mixed Number Form:

$x = 38 \frac{47}{81}$

Solve for x log base 5 of 3x+3 log base 5 of 3=5

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