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Find
lim_(h rarr0)(5log(2+h)-5log(2))/(h).
Choose 1 answer:
(A)
(5)/(2)
(B)
(5)/(2ln(10))
(C)
5log(2)
(D) The limit doesn't exist

Find $\lim _{h \rightarrow 0} \frac{5 \log (2+h)-5 \log (2)}{h}$ . $\newline$ Choose $1$ answer: $\newline$ (A) $\frac{5}{2}$ $\newline$ (B) $\frac{5}{2 \ln (10)}$ $\newline$ (C) $5 \log (2)$ $\newline$ (D) The limit doesn't exist

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Find derivatives of logarithmic functions

Full solution

Q. Find

\lim _{h \rightarrow 0} \frac{5 \log (2+h)-5 \log (2)}{h}

.

\newline

Choose

1

answer:

\newline

(A)

\frac{5}{2}

\newline

(B)

\frac{5}{2 \ln (10)}

\newline

(C)

5 \log (2)

\newline

(D) The limit doesn't exist

Combine logarithmic terms: Use the properties of logarithms to combine the terms in the numerator. $\lim_{h \rightarrow 0}\frac{5\log(2+h)-5\log(2)}{h} = \lim_{h \rightarrow 0}\frac{5\log\left(\frac{2+h}{2}\right)}{h}$
Apply constant multiple rule: Apply the constant multiple rule in limits to take the constant $5$ out of the limit. $\lim_{h \to 0}\left(\frac{5\log\left(\frac{2+h}{2}\right)}{h}\right) = 5 \times \lim_{h \to 0}\left(\frac{\log\left(\frac{2+h}{2}\right)}{h}\right)$
Use derivative definition: Use the definition of the derivative for the function $f(x) = \log(x)$ at $x = 2$ . $\lim_{h \to 0}\frac{\log((2+h)/2)}{h} = f'(2)$ where $f(x) = \log(x)$
Calculate derivative: The derivative of $\log(x)$ is $\frac{1}{x\ln(10)}$ . So, $f'(2) = \frac{1}{2\ln(10)}$ .
Multiply by constant: Multiply the derivative by the constant $5$ that we factored out earlier. $\newline$ $5 \times f'(2) = 5 \times \left(\frac{1}{2\ln(10)}\right) = \frac{5}{2\ln(10)}$

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