Porisma

Worked solution

Take logarithms when the exponential bases differ

Solve the equation 2x⋅7x=9x+12^x\cdot7^x=9^{x+1}.

CJC · O Level revision · Example 11

Answer

x=ln⁡9ln⁡14−ln⁡9≈4.97x=\dfrac{\ln9}{\ln14-\ln9}\approx4.97 (3 significant figures).

Compare your reasoning

Combine powers with the same exponent

(2×7)x=9x+1 ⟹ 14x=9x+1(2\times7)^x=9^{x+1}\ \Longrightarrow\ 14^x=9^{x+1}

The source uses (2×7)ˣ=14ˣ.

Take natural logarithms of both positive sides

ln⁡(14x)=ln⁡(9x+1) ⟹ xln⁡14=(x+1)ln⁡9\begin{aligned}&\ln(14^x)\\ & =\ln(9^{x+1})\ \Longrightarrow\ x\ln14\\ & =(x+1)\ln9\end{aligned}

Apply the printed power law to move the exponents in front.

Collect the terms containing x

x(ln⁡14−ln⁡9)=ln⁡9 ⟹ x=ln⁡9ln⁡14−ln⁡9≈4.97\begin{aligned}&x(\ln14-\ln9)\\ & =\ln9\ \Longrightarrow\ x\\ & =\frac{\ln9}{\ln14-\ln9}\approx4.97\end{aligned}

The source concludes 4.97 to 3 significant figures; the exact quotient is retained here before rounding.

How did this question go?