F: Exponentials and logarithms: Solve equations of the form a^x = b.

The idea

An equation a^x=b has the unknown in the exponent, so ordinary rearrangement cannot isolate x. The standard method takes logarithms of both sides: log(a^x)=log b, then the power law gives x log a=log b, so x=log b/log a. Any consistent base works; base 10 or e is convenient on a calculator. For example, 3^x=20 gives x=log 20/log 3≈2.73. Check by substitution. If b is a power of a, or both sides can be written with the same base, comparing indices is quicker: 2^x=8 gives x=3 because 8=2^3. Recognise when b≤0: for a>0 there is no real solution, since a^x is always positive. State answers to the accuracy required, and keep full calculator display until the final rounding to avoid premature rounding errors.

What earns the marks

• Takes logarithms of both sides of the equation, or otherwise correctly applies the power law for logarithms to bring the exponent down.
• Rearranges to x=log b/log a, or the equivalent using natural logarithms, showing the division by log a.
• Evaluates the quotient accurately, retaining sufficient figures before rounding to the accuracy requested.
• Where the equation can be written with a common base (for example b is a power of a), correctly equates the indices and solves the resulting linear equation.
• Recognises and states that no real solution exists when b is zero or negative for a positive base a.
• Checks the solution by substitution or by considering the approximate size of the answer.

Common mistakes and how to put them right

• Writing x=log b−log a instead of dividing.

  Put it right: The power law gives x log a=log b, so the correction is to divide both sides by log a.

• Taking logarithms of only one side, which breaks the equality.

  Put it right: Apply the same logarithm to both sides.

• Attempting to solve a^x=−5 and giving a real answer.

  Put it right: State that a positive base raised to any real power is positive, so there is no real solution.

• Stating the common-base condition backwards (for example 'a is a power of b').

  Put it right by noting it is b that must be a power of a, or both sides must share a base.

Exam tips

• Show the logarithmic step explicitly before evaluating, so the method is visible even if the final value is mis-keyed.
• Use the same base for both logarithms; mixing log and ln within one calculation invites error.
• Store the unrounded quotient in the calculator and round only at the end, then give the answer to the accuracy stated in the question.