 In this video we provide the solution to question number seven for practice exam four for math 1050 In which case we have to solve the exponential equation when 105 times 1 minus 10 raised to the negative 0.1 x power and that's all equal to 6.5 now the decimals might be wigging you out right now But honestly since the answer is going to be in exact form We actually don't need a calculator to help us on this one Of course, I mean you can't use it But it's not actually necessary to solve for x which is right here We have to peel the onion and get rid of all of the functions attached to x in the reverse order of operations there So we're gonna begin by dividing both sides by 105 Right if you don't and so that's gonna give you 1 minus 10 to the negative 0.1 x is equal to 6.5 over 105 if you don't like the fractions and the decimals together especially since the answer doesn't have any decimals We might want to get rid of them move the decimal over one you got to do it to the denominator as well So I give you 65 over 1,050 that fraction of course can be reduced Given that we know that 65 and 1,050 are both divisible by five five goes into 65 13 times and it goes into a 1,050 210 times like so So the next thing to do is we want to subtract one from both sides to track one subtract one now Of course on the right hand side since there's a fraction there I'm actually going to write 210 over 210 Right that then gives on the left negative 10 to the negative point one x power on the right hand side You have 13 minus 210 which is negative 197 over 210 you can times both sides by negative since there's a negative sign right there And so that gives us 10 10 to the negative point one x is equal to positive 197 over 210 at this moment. How do you get rid of the base 10? Well, you get rid of the base 10 by taking the logarithm base 10 On we're gonna so move the 10 to the right hand side becomes a logarithm Just like over here multiplication by 105 became division by 105 on the other side Positive one became negative one on the right hand side So when you move it to the other side of the equation always take the inverse operation So we end up with negative point one x is equal to log of 197 over 210 like so and so then to finish up we're gonna divide both sides by negative point one Divide by negative point one, but negative point one is actually one tenth So if you're dividing by one tenth, you're actually if you're dividing by one over ten You're actually times it by ten. So you end up with x equals negative ten log of 197 over 210 like so in which case we then see the correct answer would be f now Again, some of the arithmetic felt a little bit difficult here This is what a calculator can do to help you but be aware a calculator was not actually necessary to solve question number seven