Calculus learning

Derivative of log Base a: Change of Base and Inner Derivatives

The derivative of a logarithm depends on its base. Change of base writes log_a(x)=ln(x)/ln(a), so the natural-log derivative is scaled by a constant. The calculator’s bare log(x) means ln(x), not automatically base ten.

Direct answer

d/dx log_a(x)=1/[x ln(a)] for a>0, a≠1 and x>0.

The base is a positive constant different from one. A variable base requires a different derivative.

Derivative of log Base a worked method

For constant a, ln(a) does not depend on x. Pull 1/ln(a) outside differentiation.

Differentiate ln(x) to get 1/x, keeping the real-domain restriction x>0.

For log_a(g(x)), the chain rule gives g′(x)/[g(x)ln(a)] and requires g(x)>0.

When 0<a<1, ln(a)<0, so the logarithm and its derivative decrease. For a=1, change of base divides by zero and there is no logarithm with that base.

If the base depends on x, use the quotient rule on ln(g(x))/ln(a(x)); do not treat ln(a(x)) as a constant.

1
log_a(x)=ln(x)/ln(a)
2
(log_a(x))′=1/(x ln(a))
3
(log_a(g(x)))′=g′(x)/(g(x)ln(a))

Worked examples

d/dx log_10(x)
1/(x ln 10)

Use ln(x)/ln(10) as input.

d/dx log_2(x²+1)
2x/[(x²+1)ln 2]

The argument is always positive.

d/dx log_(1/2)(x)
1/[x ln(1/2)]

The derivative is negative for x>0.

Common mistakes

  • • Assuming all interfaces use the same log base.
  • • Using base 1.
  • • Applying the constant-base rule to log_x(x+1).

Try the supported calculator preset

The preset opens a supported expression with its variable and conditions. Teaching examples elsewhere in this guide are reference explanations, not claims that the engine supports every method.

The preset fixes the base at 10. It does not interpret the letter a as an unspecified numeric constant.

Derivatives:ln⁡xln⁡10\frac{\ln x}{\ln 10}
Open the calculator without a preset: Derivatives

References

Frequently asked questions

Does log mean ln on this site?+

Yes, the current engine treats log and ln as natural logarithms.

How do I enter another base?+

Use ln(argument)/ln(positiveConstant), such as ln(x)/ln(10).

Should log(x) get a second page separate from ln(x)?+

No. The existing natural-log guide covers that alias; this lesson covers a genuinely different base-dependent task.

Conclusion

Derivative of log Base a starts with the stated domain and conditions. Recheck those before carrying a worked example into a different problem.

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