Additional Mathematics – Logarithmic and exponential functions | e-Consult
Logarithmic and exponential functions (1 questions)
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Answer 3
Express the second term with base 3:
\[
27=3^{3}\;\;\Rightarrow\;\;27^{\,x-1}= \bigl(3^{3}\bigr)^{\,x-1}=3^{\,3(x-1)}.
\]
The equation becomes
\[
3^{\,x+2}=3^{\,3(x-1)}.
\]
Since the bases are the same, set the exponents equal:
\[
x+2=3(x-1).
\]
Solve for \(x\):
\[
x+2=3x-3\quad\Longrightarrow\quad2+3=3x-x\quad\Longrightarrow\quad5=2x\quad\Longrightarrow\quad x=\frac{5}{2}.
\]
Verification (optional):
\[
3^{\,\frac{5}{2}+2}=3^{\,\frac{9}{2}} \quad\text{and}\quad 27^{\,\frac{5}{2}-1}=27^{\,\frac{3}{2}}=(3^{3})^{\frac{3}{2}}=3^{\,\frac{9}{2}}.
\]
Both sides are equal, confirming \(x=\dfrac{5}{2}\).