$$ 5 ^ { x } , 5 ^ { x + 5 } = 29 $$

Question

UpStudy AI · Accepted Answer

1. Recognize that
   $$
   5^{x+5} = 5^x \cdot 5^5.
   $$

2. The equation
   $$
   5^{x+5} = 29
   $$
   then becomes
   $$
   5^x \cdot 5^5 = 29.
   $$

3. Solve for $$5^x$$:
   $$
   5^x = \frac{29}{5^5}.
   $$
   Since
   $$
   5^5 = 3125,
   $$
   we have
   $$
   5^x = \frac{29}{3125}.
   $$

4. To solve for $$x$$, take the logarithm of both sides. Using natural logarithms:
   $$
   \ln(5^{x+5}) = \ln(29).
   $$
   The left-hand side becomes:
   $$
   (x+5) \ln(5) = \ln(29).
   $$

5. Isolate $$x$$:
   $$
   x+5 = \frac{\ln(29)}{\ln(5)},
   $$
   so that
   $$
   x = \frac{\ln(29)}{\ln(5)} - 5.
   $$

Thus, the solutions are:
- $$5^x = \dfrac{29}{3125}$$,
- $$x = \dfrac{\ln(29)}{\ln(5)} - 5$$.
$$ x = \dfrac{\ln(29)}{\ln(5)} - 5 $$

\( 5 ^ { x } , 5 ^ { x + 5 } = 29 \)

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