The log ⁡ a x \log_{a} x Goes To The h h

Calculus Level 2

What is the derivative f ′ ( x ) f'(x) of f ( x ) = log ⁡ a x f(x)=\log_{a} x ?

1 x ln ⁡ ( a ) , x > 0 \frac{1}{x\ln(a)},\ x>0 − 1 a ln ⁡ ( x ) , x > 0 -\frac{1}{a\ln(x)},\ x>0 1 x log ⁡ ( a ) , x > 0 \frac{1}{x\log(a)},\ x>0 1 x ln ⁡ ( a ) , x ≥ 0 \frac{1}{x\ln(a)},\ x\geq0 − 1 a ln ⁡ ( x ) , x ≥ 0 -\frac{1}{a\ln(x)},\ x\geq0

Andrei Li by Andrei Li , 16, Canada

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2 solutions

Chew-Seong Cheong
Jul 21, 2018

f ( x ) = log ⁡ a x ⟹ f ′ ( x ) = d d x log ⁡ a x = d d x ( ln ⁡ x ln ⁡ a ) = 1 x ln ⁡ a for x > 0 \begin{aligned} f(x) & = \log_a x \\ \implies f'(x) & = \frac d{dx} \log_a x \\ & = \frac d{dx}\left(\frac {\ln x}{\ln a}\right) \\ & = \boxed{\dfrac 1{x\ln a}} & \small \color{#3D99F6} \text{for }x > 0 \end{aligned}

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X X
Jul 21, 2018

log ⁡ a x = ln ⁡ x ln ⁡ a \log_a{x}=\dfrac{\ln{x}}{\ln a} ,so the derivative is 1 x × 1 ln ⁡ a = 1 x ln ⁡ a \dfrac1x\times\dfrac1{\ln a}=\dfrac1{x\ln a}

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