Use logarithms to solve the equation 2^5x = 3^2x+1 , giving the answer correct to 3 significant figures.

Firstly, I would make sure that the student is aware of the basic conept of logarithims and run through the basic "laws" of logarithims to make sure that they have the knowledge they need to answer this type of question. Then I would give them a moment to see if they can spot which law (or laws) can be applied to this equation (bearing in mind there are multiple combinations that can be used). Initially, the logical thing to do is to apply to natural log to the equation(on both sides, needless to say), to come to the equation ln2^5x=ln3^2x+1 . After this, you apply the law 'lnx^y = ylnx' to both sides. This will lead to the equation being written as: 5xln2=(2x+1)ln3. From here on, it is a simple case of algebra, where the numbers 'ln2' and 'ln3' are treated as constants (an idea which may seem uncomfortable at first, and leads to a few silly mistakes like treating the symbol "ln" as a constant). The answer is 0.866. It is subtly indicated in the question that the answer is likely not to be "pretty" as you are askd to round to three significant figures.