Solve 32log3(x - 5) = 49 for x.

Step-by-step solution

1. 1

   State the domain before solving. A logarithm needs a strictly positive argument, so

   $x-5>0\qquad\Longrightarrow\qquad x>5$

   Any candidate answer must clear this bar; writing the restriction down first is what makes the final check meaningful rather than an afterthought.

2. 2

   Isolate the logarithm. Divide both sides by the coefficient $32$:

   $\log_3(x-5)=\frac{49}{32}$

   Leaving the right side as the exact fraction $\tfrac{49}{32}=1.53125$ keeps the answer exact.

3. 3

   Convert to exponential form. The defining equivalence for logarithms is

   $\log_b(A)=c\iff A=b^{\,c}$

   Here $b=3$, $A=x-5$, $c=\tfrac{49}{32}$, so

   $x-5=3^{49/32}$

4. 4

   Solve for x. Add $5$:

   $x=5+3^{49/32}$

   This is the exact answer; no further simplification is possible because $\tfrac{49}{32}$ is not an integer and $49$ and $32$ share no factor with $3$.

5. 5

   Evaluate numerically and confirm the domain. Since $3^{49/32}=3^{1.53125}=5.37764$,

   $x=5+5.37764=10.37764$

   As a size check, $3^{1.5}=\sqrt{27}\approx5.196$ and $3^{1.53125}$ should be a little larger — it is ✓. And $x\approx10.378>5$, so the logarithm is defined and the solution is valid.

6. 6

   Verify by substituting back. With $x=10.3776424$: $x-5=5.3776424$, and $\log_3(5.3776424)=\dfrac{\ln 5.3776424}{\ln 3}=1.531250$, so $32\times1.53125=49.0000$ ✓.