Please Answer 6 a and b as well as 8 a and b 8 For the funct

Please Answer 6 a. and b. as well as 8 a. and b.

8. For the functions f(x)=ex and g(x)=ln3x b, (g of)(x) (simplify using rules for logarithms and In3s 1.0986) Page 4 of 7

Solution

8)

f(x) = e^x

g(x) = ln 3x

a)

fog(x) = e^{ln 3x}     = 3x

b)

gof(x) = ln 3e^x                      // we just replaced \"x\" by e^x

= ln(3) + ln(e^x)                    //log(mn) = log(m) + log(n) ,Multiplication inside the log can be turned into addition outside the log

= ln(3) + x.ln(e)                    // log(mⁿ) = n · log(m)      , An exponent on everything inside a log can be moved out front as a multiplier

=ln(3) + x                   //ln(e) = 1

gof(x) = ln(3) +x

.

6

pH = - log[H⁺]

a)

given

             [H⁺] = 8 X 10^-9

now we subtituite this in our pH equation

pH = - log [H⁺]

pH = - log[8 x 10^-9]

pH = - [ log(8) + log(10^-9) ]    //log(mn) = log(m) + log(n) ,Multiplication inside the log can be turned into addition outside the log

pH= - log(8) - log(10^-9)

pH = -log(8) + 9 log(10)              //   \"-9\" is moved out as a multiplier

pH = -log(2³) + 9           // log(10) = 1          we can write 8 as 2³

pH = -3log(2) + 9

pH = -3 *(0.3010) + 9       // given log(2) = 0.3010

pH = 8.097

b)

pH= 7.45

7.45 = - log [H⁺]

-7.45 = log [H⁺]              // log means log with base 10 = log₁₀

-7.45 = log₁₀ [H⁺]

10^-7.45 = H⁺            // log_a b = c      => b = a^c

H⁺ = 10^-7.45                         // if there is a negative exponent then the reciprocal of that will give positive exponent

H⁺ = 1/10^7.45               

H⁺ = 3.54 X 10^-8