Solve for x 5^(x-3)=137

$5^{x - 3} = 137$

Step 1

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

$\ln (5^{x - 3}) = \ln (137)$

Step 2

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

$(x - 3) \ln (5) = \ln (137)$

Step 3

Simplify the left side.

Apply the distributive property.

$x \ln (5) - 3 \ln (5) = \ln (137)$

Step 4

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

$x \ln (5) - 3 \ln (5) - \ln (137) = 0$

Step 5

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

Add $3 \ln (5)$ to both sides of the equation.

$x \ln (5) - \ln (137) = 3 \ln (5)$

Add $\ln (137)$ to both sides of the equation.

$x \ln (5) = 3 \ln (5) + \ln (137)$

Step 6

Divide each term in $x \ln (5) = 3 \ln (5) + \ln (137)$ by $\ln (5)$ and simplify.

Divide each term in $x \ln (5) = 3 \ln (5) + \ln (137)$ by $\ln (5)$ .

$\frac{x \ln (5)}{\ln (5)} = \frac{3 \ln (5)}{\ln (5)} + \frac{\ln (137)}{\ln (5)}$

Simplify the left side.

Cancel the common factor of $\ln (5)$ .

Cancel the common factor.

$\frac{x \ln (5)}{\ln (5)} = \frac{3 \ln (5)}{\ln (5)} + \frac{\ln (137)}{\ln (5)}$

Divide $x$ by $1$ .

$x = \frac{3 \ln (5)}{\ln (5)} + \frac{\ln (137)}{\ln (5)}$

Simplify the right side.

Cancel the common factor of $\ln (5)$ .

Cancel the common factor.

$x = \frac{3 \ln (5)}{\ln (5)} + \frac{\ln (137)}{\ln (5)}$

Divide $3$ by $1$ .

$x = 3 + \frac{\ln (137)}{\ln (5)}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$x = 3 + \frac{\ln (137)}{\ln (5)}$

Decimal Form:

$x = 6.05695602 \dots$