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$The functions f and g are given by {:[f(x)=(1)/(2)log_(10)(x-3)],[g(x)=3ln(x+2)]:} (A) Solve f(x)=1 for values of x in the domain of f.$

The functions $f$ and $g$ are given by $\newline$ $\begin{array}{l} f(x)=\frac{1}{2} \log _{10}(x-3) \\ g(x)=3 \ln (x+2) \end{array}$ $\newline$ (A) Solve $f(x)=1$ for values of $x$ in the domain of $f$ .

Full solution

Q. The functions

f

and

g

are given by

\newline

\begin{array}{l} f(x)=\frac{1}{2} \log _{10}(x-3) \\ g(x)=3 \ln (x+2) \end{array}

\newline

(A) Solve

f(x)=1

for values of

x

in the domain of

f

Set $f(x)$ equal to $1$ : Set the function $f(x)$ equal to $1$ and solve for x. $\newline$ $f(x) = \frac{1}{2}\log_{10}(x-3) = 1$
Multiply by $2$ : Multiply both sides of the equation by $2$ to isolate the logarithm. $\newline$ $2 \times (\frac{1}{2})\log_{10}(x-3) = 2 \times 1$ $\newline$ $\log_{10}(x-3) = 2$
Convert to exponential form: Convert the logarithmic equation to its exponential form. $\newline$ $10^{\log_{10}(x-3)} = 10^2$ $\newline$ $x - 3 = 100$
Add $3$ to solve for x: Add $3$ to both sides of the equation to solve for x. $\newline$ $x - 3 + 3 = 100 + 3$ $\newline$ $x = 103$