What is the derivative of log base 2 of x?
Understand the Problem
The question is asking for the derivative of the function log(x) with base 2. We will use the differentiation rules to find it, particularly the change of base formula for logarithms.
Answer
The derivative of $\log_2(x)$ is $f'(x) = \frac{1}{x \ln(2)}$.
Answer for screen readers
The derivative of the function $\log_2(x)$ is given by: $$ f'(x) = \frac{1}{x \ln(2)} $$
Steps to Solve
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Identify the function The function we are taking the derivative of is $f(x) = \log_2(x)$.
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Apply change of base formula We can use the change of base formula for logarithms which states that: $$ \log_b(x) = \frac{\log_k(x)}{\log_k(b)} $$ In our case, let's convert $\log_2(x)$ to the natural logarithm (base $e$): $$ \log_2(x) = \frac{\ln(x)}{\ln(2)} $$
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Differentiate the new expression Now we need to take the derivative of $f(x) = \frac{\ln(x)}{\ln(2)}$. Since $\ln(2)$ is a constant, we can use the constant multiple rule: $$ f'(x) = \frac{1}{\ln(2)} \cdot \frac{d}{dx}(\ln(x)) $$
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Calculate the derivative of ln(x) We know that the derivative of $\ln(x)$ is $\frac{1}{x}$: $$ \frac{d}{dx}(\ln(x)) = \frac{1}{x} $$
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Combine the results Thus, we have: $$ f'(x) = \frac{1}{\ln(2)} \cdot \frac{1}{x} $$
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Final result This gives us: $$ f'(x) = \frac{1}{x \ln(2)} $$
The derivative of the function $\log_2(x)$ is given by: $$ f'(x) = \frac{1}{x \ln(2)} $$
More Information
The derivative tells us how the function $\log_2(x)$ changes with respect to $x$. The base of the logarithm (in this case, 2) impacts the value of the derivative because it affects the constant of proportionality $\ln(2)$.
Tips
- Misapplying the change of base formula: Remember to convert the base correctly and keep track of constants.
- Forgetting the derivative of $\ln(x)$: It's easy to overlook that this derivative is $\frac{1}{x}$.
