Question

Please Need help on these

Answer 1

                         Question # 3

Answer:

$x = 13.094$

Step-by-Step Explanation

Considering the exponential equation

$e^{x-9}-6=54$

Solving the exponentiation equation

$e^{x-9}-6=54$

$\mathrm{Add\:}6\mathrm{\:to\:both\:sides}$

$e^{x-9}-6+6=54+6$

$e^{x-9}=60$

$\mathrm{If\:}f\left(x\right)=g\left(x\right)\mathrm{,\:then\:}\ln \left(f\left(x\right)\right)=\ln \left(g\left(x\right)\right)$

$\ln \left(e^{x-9}\right)=\ln \left(60\right)$

$\mathrm{Apply\:log\:rule}:\quad \log _a\left(x^b\right)=b\cdot \log _a\left(x\right)$

$\ln \left(e^{x-9}\right)=\left(x-9\right)\ln \left(e\right)$

$\left(x-9\right)\ln \left(e\right)=\ln \left(60\right)$

$\mathrm{Apply\:log\:rule}:\quad \log _a\left(a\right)=1$

$\ln \left(e\right)=1$

$x-9=\ln \left(60\right)$

$x=\ln \left(60\right)+9$

$x=13.094$          ∵ $\ln \left(60\right)=4.094$

Therefore, x = 13.094

                      Question # 4

Answer:

$x=0.386$

Step-by-Step Explanation

Considering the exponential equation

$3e^{9x}-6=91$

Solving the exponentiation equation

$3e^{9x}-6=91$

$\mathrm{Add\:}6\mathrm{\:to\:both\:sides}$

$3e^{9x}=97$

$\mathrm{Divide\:both\:sides\:by\:}3$

$\frac{3e^{9x}}{3}=\frac{97}{3}$

$e^{9x}=\frac{97}{3}$

$\mathrm{If\:}f\left(x\right)=g\left(x\right)\mathrm{,\:then\:}\ln \left(f\left(x\right)\right)=\ln \left(g\left(x\right)\right)$

$\ln \left(e^{9x}\right)=9x\ln \left(e\right)$

$9x\ln \left(e\right)=\ln \left(\frac{97}{3}\right)$

$\mathrm{Apply\:log\:rule}:\quad \log _a\left(a\right)=1$

$\ln \left(e\right)=1$

$9x=\ln \left(\frac{97}{3}\right)$

$\mathrm{Divide\:both\:sides\:by\:}9$

$\frac{9x}{9}=\frac{\ln \left(\frac{97}{3}\right)}{9}$

$x=\frac{\ln \left(\frac{97}{3}\right)}{9}$

$x=\frac{3.476}{9}$       ∵ $ln\left(\frac{97}{3}\right)=3.476$

$x=0.386$    

Therefore, x = 0.386