Logarithms & Exponentials

Topic overview

The key to exponential and logarithm questions is not computation, but first spotting that you can use a common base, make a substitution, or take logarithms to reduce the order of the non-linear structure.

Key concepts

If $a^x$ , $a^{-x}$ or $a^{2x}$ appear simultaneously, let $u=a^x$ , and record $u>0$ . Solve the equation in $u$ and then substitute back to $x$ ; negative values of $u$ must be discarded.

Key points

◆State that the substitution variable is positive
◆The argument of a logarithm must be positive
◆Give numerical solutions to the required number of significant figures

Worked examples

Try each example yourself before checking the explanation.

1Two-way exponentiation

Solve $2^{x+1}+8\cdot2^{-x}=10$ .

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