Answer:
[tex]x=25.24[/tex]
Step-by-step explanation:
To find the value of [tex]x[/tex], we are solving the exponential equation [tex]-6.2^{0.1x} =-100[/tex] using logarithms.
Let's solve it step-by-step
Step 1. Divide both sides of the equation by -1 to get rid of the negative signs:
[tex]-6.2^{0.1x} =-100[/tex]
[tex]\frac{-6.2^{0.1x}}{-1} =\frac{-100}{-1}[/tex]
[tex]6.2^{0.1x} =100[/tex]
Step 2. Take natural logarithm to bot sides of the equation:
[tex]ln(6.2^{0.1x)}=ln(100)[/tex]
Step 3. Use the power rule for logarithms: [tex]ln(a^{x} )=xln(a)[/tex]
For our equation: [tex]a=6.2[/tex] and [tex]x=0.1x[/tex]
[tex]ln(6.2^{0.1x)}=ln(100)[/tex]
[tex]0.1xln(6.2)=ln(100)[/tex]
Step 4. Divide both sides of the equation by [tex]0.1ln(6.2)[/tex]
[tex]0.1xln(6.2)=ln(100)[/tex]
[tex]\frac{0.1xln(6.2)}{0.1ln(6.2)} =\frac{ln(100)}{0.1ln(6.2)}[/tex]
[tex]x=\frac{ln(100)}{0.1ln(6.2)}[/tex]
[tex]x=25.24[/tex]
We can conclude that the value of x in our exponential equation is approximately 25.24.
