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MathematicsA LevelExponentials & Logarithms

Power & change of base formula

The power & change of base formula as it is used in A Level Mathematics, taken from the A Level Mathematics formula sheet.

The formula

log⁡axk=klog⁡ax,log⁡ax=log⁡bxlog⁡ba\log_a x^k = k\log_a x, \quad \log_a x = \dfrac{\log_b x}{\log_b a}

How to use it in the exam

  • Write the formula down before substituting — most mark schemes award a method mark for a correct, clearly quoted formula.
  • Check every quantity is in consistent units before you substitute, then substitute in one step so the working stays easy to follow.
  • Show the rearrangement rather than jumping to the answer, and give the final value to the accuracy the question asks for.

Related exponentials & logarithms formulas

Log laws
log⁡axy=log⁡ax+log⁡ay,log⁡axy=log⁡ax−log⁡ay\log_a xy = \log_a x + \log_a y, \quad \log_a \dfrac{x}{y} = \log_a x - \log_a y
Natural log relationship
y=ex  ⟺  x=ln⁡yy = e^{x} \iff x = \ln y

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