Solve for r (2/3)^(r-8)=7/4

${(\frac{2}{3})}^{r - 8} = \frac{7}{4}$

Step 1

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

$\ln ({(\frac{2}{3})}^{r - 8}) = \ln (\frac{7}{4})$

Step 2

Expand the left side.

Expand $\ln ({(\frac{2}{3})}^{r - 8})$ by moving $r - 8$ outside the logarithm.

$(r - 8) \ln (\frac{2}{3}) = \ln (\frac{7}{4})$

Rewrite $\ln (\frac{2}{3})$ as $\ln (2) - \ln (3)$ .

$(r - 8) (\ln (2) - \ln (3)) = \ln (\frac{7}{4})$

Step 3

Simplify the left side.

Simplify $(r - 8) (\ln (2) - \ln (3))$ .

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

$(r - 8) \ln (\frac{2}{3}) = \ln (\frac{7}{4})$

Apply the distributive property.

$r \ln (\frac{2}{3}) - 8 \ln (\frac{2}{3}) = \ln (\frac{7}{4})$

Step 4

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

$r \ln (\frac{2}{3}) - 8 \ln (\frac{2}{3}) - \ln (\frac{7}{4}) = 0$

Step 5

Move all terms not containing $r$ to the right side of the equation.

Add $8 \ln (\frac{2}{3})$ to both sides of the equation.

$r \ln (\frac{2}{3}) - \ln (\frac{7}{4}) = 8 \ln (\frac{2}{3})$

Add $\ln (\frac{7}{4})$ to both sides of the equation.

$r \ln (\frac{2}{3}) = 8 \ln (\frac{2}{3}) + \ln (\frac{7}{4})$

Step 6

Divide each term in $r \ln (\frac{2}{3}) = 8 \ln (\frac{2}{3}) + \ln (\frac{7}{4})$ by $\ln (\frac{2}{3})$ and simplify.

Divide each term in $r \ln (\frac{2}{3}) = 8 \ln (\frac{2}{3}) + \ln (\frac{7}{4})$ by $\ln (\frac{2}{3})$ .

$\frac{r \ln (\frac{2}{3})}{\ln (\frac{2}{3})} = \frac{8 \ln (\frac{2}{3})}{\ln (\frac{2}{3})} + \frac{\ln (\frac{7}{4})}{\ln (\frac{2}{3})}$

Simplify the left side.

Cancel the common factor of $\ln (\frac{2}{3})$ .

Cancel the common factor.

$\frac{r \ln (\frac{2}{3})}{\ln (\frac{2}{3})} = \frac{8 \ln (\frac{2}{3})}{\ln (\frac{2}{3})} + \frac{\ln (\frac{7}{4})}{\ln (\frac{2}{3})}$

Divide $r$ by $1$ .

$r = \frac{8 \ln (\frac{2}{3})}{\ln (\frac{2}{3})} + \frac{\ln (\frac{7}{4})}{\ln (\frac{2}{3})}$

Simplify the right side.

Cancel the common factor of $\ln (\frac{2}{3})$ .

Cancel the common factor.

$r = \frac{8 \ln (\frac{2}{3})}{\ln (\frac{2}{3})} + \frac{\ln (\frac{7}{4})}{\ln (\frac{2}{3})}$

Divide $8$ by $1$ .

$r = 8 + \frac{\ln (\frac{7}{4})}{\ln (\frac{2}{3})}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$r = 8 + \frac{\ln (\frac{7}{4})}{\ln (\frac{2}{3})}$

Decimal Form:

$r = 6.61981764 \dots$

Solve for r (2/3)^(r-8)=7/4

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