Solve 5^(x+2) = 3^(5-x) for x.

Step-by-step solution

1. 1

   See why logarithms are unavoidable. The bases $5$ and $3$ share no common power, so the exponents cannot simply be equated. Taking a logarithm of both sides is the standard move — and any base works, since it cancels out of the final ratio.

2. 2

   Take natural logs and bring the exponents down. Using $\ln(A^k)=k\ln A$:

   $(x+2)\ln5=(5-x)\ln3$

   The equation is now linear in $x$, with $\ln5$ and $\ln3$ acting as ordinary constants.

3. 3

   Expand and gather the x terms.

   $x\ln5+2\ln5=5\ln3-x\ln3\qquad\Longrightarrow\qquad x\ln5+x\ln3=5\ln3-2\ln5$

   Both $x$ terms move to the left, which keeps the coefficient positive.

4. 4

   Factor and use the product rule for logs.

   $x(\ln5+\ln3)=5\ln3-2\ln5$

   Since $\ln5+\ln3=\ln(5\cdot3)=\ln15$, this simplifies to

   $x=\frac{5\ln3-2\ln5}{\ln15}$

5. 5

   Evaluate, and correct a widely quoted wrong decimal. With $\ln3=1.098612$, $\ln5=1.609438$ and $\ln15=2.708050$:

   $x=\frac{5(1.098612)-2(1.609438)}{2.708050}=\frac{5.493061-3.218876}{2.708050}=\frac{2.274185}{2.708050}=0.839787$

   So $x\approx0.8398$. A value of $1.037$ is sometimes given for this equation, but it is wrong — see the check below.

6. 6

   Verify by substitution. With $x=0.8397871$: $5^{x+2}=5^{2.8397871}=96.5886$ and $3^{5-x}=3^{4.1602129}=96.5886$ — equal to seven significant figures ✓. Testing the incorrect $x=1.037$ instead gives $5^{3.037}=132.67$ against $3^{3.963}=77.77$, which are not remotely equal ✗.