Full solution

Q. (A) The functions

g

and

h

are given by

\newline

\begin{array}{l} g(x)=\log _{4}(2 x) \\ h(x)=\frac{\left(e^{x}\right)^{5}}{e^{1 / 4}} \end{array}

\newline

(i) Solve

g(x)=3

for values of

x

in the domain of

g

\newline

(ii) Solve

h(x)=e^{1 / 2}

for values of

x

in the domain of

h

Solve $g(x)=3$ : Solve $g(x)=3$ for values of $x$ in the domain of $g$ .
$g(x) = \log_4(2x)$
Set $g(x)$ to $3$ and solve for $x$ .
$\log_4(2x) = 3$
Convert to exponential: Convert the logarithmic equation to an exponential equation. $\newline$ $4^3 = 2x$
Calculate value of $4^3$ : Calculate the value of $4^3$ . $\newline$ $4^3 = 64$
Solve for x: Solve for x. $\newline$ $64 = 2x$ $\newline$ $x = \frac{64}{2}$ $\newline$ $x = 32$
Solve $h(x)=e^{\frac{1}{2}}$ : Solve $h(x)=e^{\frac{1}{2}}$ for values of $x$ in the domain of $h$ .
$h(x) = \frac{(e^x)^5}{e^{\frac{1}{4}}}$
Set $h(x)$ to $e^{\frac{1}{2}}$ and solve for $x$ .
$\frac{(e^x)^5}{e^{\frac{1}{4}}} = e^{\frac{1}{2}}$
Multiply by $e^{\frac{1}{4}}$ : Multiply both sides by $e^{\frac{1}{4}}$ to isolate $(e^x)^5$ . $\newline$ $(e^x)^5 = e^{\frac{1}{2}} \cdot e^{\frac{1}{4}}$
Combine exponents: Combine the exponents on the right side using the property $e^{a+b} = e^a \cdot e^b$ . $\newline$ $(e^x)^5 = e^{\frac{1}{2} + \frac{1}{4}}$
Calculate sum of exponents: Calculate the sum of the exponents on the right side. $\newline$ $\frac{1}{2} + \frac{1}{4} = \frac{2}{4} + \frac{1}{4} = \frac{3}{4}$
Substitute sum: Substitute the sum back into the equation. $\newline$ $(e^x)^5 = e^{\frac{3}{4}}$
Take fifth root: Take the fifth root of both sides to solve for $e^x$ . $\newline$ $e^x = (e^{\frac{3}{4}})^{\frac{1}{5}}$
Calculate exponent: Calculate the exponent on the right side using the property $(e^a)^{1/b} = e^{a/b}$ . $\newline$ $e^x = e^{\frac{3}{4} \cdot \frac{1}{5}}$
Calculate product: Calculate the product of the exponents on the right side. $\frac{3}{4} \times \frac{1}{5} = \frac{3}{20}$
Substitute product: Substitute the product back into the equation. $\newline$ $e^x = e^{\frac{3}{20}}$
Equate exponents: Since the bases are the same and the equation is of the form $e^x = e^y$ , we can equate the exponents. $\newline$ $x = \frac{3}{20}$