Question # 3
Answer:
[tex]x = 13.094[/tex]
Step-by-Step Explanation
Considering the exponential equation
[tex]e^{x-9}-6=54[/tex]
Solving the exponentiation equation
[tex]e^{x-9}-6=54[/tex]
[tex]\mathrm{Add\:}6\mathrm{\:to\:both\:sides}[/tex]
[tex]e^{x-9}-6+6=54+6[/tex]
[tex]e^{x-9}=60[/tex]
[tex]\mathrm{If\:}f\left(x\right)=g\left(x\right)\mathrm{,\:then\:}\ln \left(f\left(x\right)\right)=\ln \left(g\left(x\right)\right)[/tex]
[tex]\ln \left(e^{x-9}\right)=\ln \left(60\right)[/tex]
[tex]\mathrm{Apply\:log\:rule}:\quad \log _a\left(x^b\right)=b\cdot \log _a\left(x\right)[/tex]
[tex]\ln \left(e^{x-9}\right)=\left(x-9\right)\ln \left(e\right)[/tex]
[tex]\left(x-9\right)\ln \left(e\right)=\ln \left(60\right)[/tex]
[tex]\mathrm{Apply\:log\:rule}:\quad \log _a\left(a\right)=1[/tex]
[tex]\ln \left(e\right)=1[/tex]
[tex]x-9=\ln \left(60\right)[/tex]
[tex]x=\ln \left(60\right)+9[/tex]
[tex]x=13.094[/tex] ∵ [tex]\ln \left(60\right)=4.094[/tex]
Therefore, x = 13.094
Question # 4
Answer:
[tex]x=0.386[/tex]
Step-by-Step Explanation
Considering the exponential equation
[tex]3e^{9x}-6=91[/tex]
Solving the exponentiation equation
[tex]3e^{9x}-6=91[/tex]
[tex]\mathrm{Add\:}6\mathrm{\:to\:both\:sides}[/tex]
[tex]3e^{9x}=97[/tex]
[tex]\mathrm{Divide\:both\:sides\:by\:}3[/tex]
[tex]\frac{3e^{9x}}{3}=\frac{97}{3}[/tex]
[tex]e^{9x}=\frac{97}{3}[/tex]
[tex]\mathrm{If\:}f\left(x\right)=g\left(x\right)\mathrm{,\:then\:}\ln \left(f\left(x\right)\right)=\ln \left(g\left(x\right)\right)[/tex]
[tex]\ln \left(e^{9x}\right)=9x\ln \left(e\right)[/tex]
[tex]9x\ln \left(e\right)=\ln \left(\frac{97}{3}\right)[/tex]
[tex]\mathrm{Apply\:log\:rule}:\quad \log _a\left(a\right)=1[/tex]
[tex]\ln \left(e\right)=1[/tex]
[tex]9x=\ln \left(\frac{97}{3}\right)[/tex]
[tex]\mathrm{Divide\:both\:sides\:by\:}9[/tex]
[tex]\frac{9x}{9}=\frac{\ln \left(\frac{97}{3}\right)}{9}[/tex]
[tex]x=\frac{\ln \left(\frac{97}{3}\right)}{9}[/tex]
[tex]x=\frac{3.476}{9}[/tex] ∵ [tex]ln\left(\frac{97}{3}\right)=3.476[/tex]
[tex]x=0.386[/tex]
Therefore, x = 0.386
