Solve for x (3/5)^(x+1)=25/9

${(\frac{3}{5})}^{x + 1} = \frac{25}{9}$

Step 1

Take the natural logarithm of both sides of the equation to remove the variable from the exponent.

$\ln ({(\frac{3}{5})}^{x + 1}) = \ln (\frac{25}{9})$

Step 2

Expand the left side.

Expand $\ln ({(\frac{3}{5})}^{x + 1})$ by moving $x + 1$ outside the logarithm.

$(x + 1) \ln (\frac{3}{5}) = \ln (\frac{25}{9})$

Rewrite $\ln (\frac{3}{5})$ as $\ln (3) - \ln (5)$ .

$(x + 1) (\ln (3) - \ln (5)) = \ln (\frac{25}{9})$

Step 3

Simplify the left side.

Simplify $(x + 1) (\ln (3) - \ln (5))$ .

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

$(x + 1) \ln (\frac{3}{5}) = \ln (\frac{25}{9})$

Apply the distributive property.

$x \ln (\frac{3}{5}) + 1 \ln (\frac{3}{5}) = \ln (\frac{25}{9})$

Multiply $\ln (\frac{3}{5})$ by $1$ .

$x \ln (\frac{3}{5}) + \ln (\frac{3}{5}) = \ln (\frac{25}{9})$

Step 4

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

$x \ln (\frac{3}{5}) + \ln (\frac{3}{5}) - \ln (\frac{25}{9}) = 0$

Step 5

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

$x \ln (\frac{3}{5}) + \ln (\frac{\frac{3}{5}}{\frac{25}{9}}) = 0$

Step 6

Simplify each term.

Multiply the numerator by the reciprocal of the denominator.

$x \ln (\frac{3}{5}) + \ln (\frac{3}{5} \cdot \frac{9}{25}) = 0$

Multiply $\frac{3}{5} \cdot \frac{9}{25}$ .

Multiply $\frac{3}{5}$ by $\frac{9}{25}$ .

$x \ln (\frac{3}{5}) + \ln (\frac{3 \cdot 9}{5 \cdot 25}) = 0$

Multiply $3$ by $9$ .

$x \ln (\frac{3}{5}) + \ln (\frac{27}{5 \cdot 25}) = 0$

Multiply $5$ by $25$ .

$x \ln (\frac{3}{5}) + \ln (\frac{27}{125}) = 0$

Step 7

Subtract $\ln (\frac{27}{125})$ from both sides of the equation.

$x \ln (\frac{3}{5}) = - \ln (\frac{27}{125})$

Step 8

Divide each term in $x \ln (\frac{3}{5}) = - \ln (\frac{27}{125})$ by $\ln (\frac{3}{5})$ and simplify.

Divide each term in $x \ln (\frac{3}{5}) = - \ln (\frac{27}{125})$ by $\ln (\frac{3}{5})$ .

$\frac{x \ln (\frac{3}{5})}{\ln (\frac{3}{5})} = \frac{- \ln (\frac{27}{125})}{\ln (\frac{3}{5})}$

Simplify the left side.

Cancel the common factor of $\ln (\frac{3}{5})$ .

Cancel the common factor.

$\frac{x \ln (\frac{3}{5})}{\ln (\frac{3}{5})} = \frac{- \ln (\frac{27}{125})}{\ln (\frac{3}{5})}$

Divide $x$ by $1$ .

$x = \frac{- \ln (\frac{27}{125})}{\ln (\frac{3}{5})}$

Simplify the right side.

Move the negative in front of the fraction.

$x = - \frac{\ln (\frac{27}{125})}{\ln (\frac{3}{5})}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{\ln (\frac{27}{125})}{\ln (\frac{3}{5})}$

Decimal Form:

$x = - 3$

Solve for x (3/5)^(x+1)=25/9

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