Question

first use the appropriate properties of logarithms to rewrite f(x), and then find f(x). f(x)=10x + ln 10x
rewrite f(x) using properties of logarithms.
f(x)= (do not simplify.)

Explanation:

Step1: Apply log - product rule

Recall the property $\ln(ab)=\ln a+\ln b$. For $\ln(10x)$, we have $\ln(10x)=\ln10+\ln x$. So $f(x)=10x+\ln10+\ln x$.

Step2: Differentiate term - by - term

The derivative of $10x$ with respect to $x$ is $10$ (using the power rule $\frac{d}{dx}(ax)=a$ where $a = 10$), the derivative of a constant $\ln10$ with respect to $x$ is $0$, and the derivative of $\ln x$ with respect to $x$ is $\frac{1}{x}$. So $f^{\prime}(x)=10 + 0+\frac{1}{x}=10+\frac{1}{x}$.

Answer:

Rewrite $f(x)$: $f(x)=10x+\ln10+\ln x$
$f^{\prime}(x)=10+\frac{1}{x}$