4.5 Exponential and Logarithmic Equations

Solve $3^{x^{2}}=3^{(3)x-2}$. Give every real solution exactly and to six decimal places when needed, and check the original equation. Written work is required; selecting an answer alone is not a complete response. Complete the written work or drawing first. The selected answer checks the result only; your reasoning or drawing needs teacher/self-review.

Equate exponents by injectivity: $x^{2} - 3 x + 2$ equals zero.
Factor: $(x-1)(x-2)=0$.
Both roots $1$, $2$ are valid because exponentials have all-real input domain.
At each root the two exponents agree; the original sides therefore agree.
The solutions are $x=1\approx1.000000$, $x=2\approx2.000000$. These decimals are rounded to six places; the preceding logarithmic values are exact.
Written-work review:

• Accept mathematically equivalent derivations and valid alternative counterexamples. Every requested conclusion, original-domain restriction, justification, precision and unit requirement still applies; the listed solution is one acceptable route.
• Required written evidence: Equate exponents by injectivity: $x^{2} - 3 x + 2$ equals zero.
• Required written evidence: Factor: $(x-1)(x-2)=0$.
• Required written evidence: Both roots $1$, $2$ are valid because exponentials have all-real input domain.
• Required written evidence: At each root the two exponents agree; the original sides therefore agree.
• Required written evidence: The solutions are $x=1\approx1.000000$, $x=2\approx2.000000$. These decimals are rounded to six places; the preceding logarithmic values are exact.